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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ -->
<title>Math/Combinatorics/CombinatorialHopfAlgebra.hs</title>
<link type='text/css' rel='stylesheet' href='hscolour.css' />
</head>
<body>
<pre><a name="line-1"></a><span class='hs-comment'>-- Copyright (c) 2012-2015, David Amos. All rights reserved.</span>
<a name="line-2"></a>
<a name="line-3"></a><span class='hs-comment'>{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, NoMonomorphismRestriction, ScopedTypeVariables, DeriveFunctor #-}</span>
<a name="line-4"></a>
<a name="line-5"></a><span class='hs-comment'>-- |A module defining the following Combinatorial Hopf Algebras, together with coalgebra or Hopf algebra morphisms between them:</span>
<a name="line-6"></a><span class='hs-comment'>--</span>
<a name="line-7"></a><span class='hs-comment'>-- * Sh, the Shuffle Hopf algebra</span>
<a name="line-8"></a><span class='hs-comment'>--</span>
<a name="line-9"></a><span class='hs-comment'>-- * SSym, the Malvenuto-Reutnenauer Hopf algebra of permutations</span>
<a name="line-10"></a><span class='hs-comment'>--</span>
<a name="line-11"></a><span class='hs-comment'>-- * YSym, the (dual of the) Loday-Ronco Hopf algebra of binary trees</span>
<a name="line-12"></a><span class='hs-comment'>--</span>
<a name="line-13"></a><span class='hs-comment'>-- * QSym, the Hopf algebra of quasi-symmetric functions (having a basis indexed by compositions)</span>
<a name="line-14"></a><span class='hs-comment'>--</span>
<a name="line-15"></a><span class='hs-comment'>-- * Sym, the Hopf algebra of symmetric functions (having a basis indexed by integer partitions)</span>
<a name="line-16"></a><span class='hs-comment'>--</span>
<a name="line-17"></a><span class='hs-comment'>-- * NSym, the Hopf algebra of non-commutative symmetric functions</span>
<a name="line-18"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Combinatorics</span><span class='hs-varop'>.</span><span class='hs-conid'>CombinatorialHopfAlgebra</span> <span class='hs-keyword'>where</span>
<a name="line-19"></a>
<a name="line-20"></a><span class='hs-comment'>-- Sources:</span>
<a name="line-21"></a>
<a name="line-22"></a><span class='hs-comment'>-- Structure of the Malvenuto-Reutenauer Hopf algebra of permutations</span>
<a name="line-23"></a><span class='hs-comment'>-- Marcelo Aguiar and Frank Sottile</span>
<a name="line-24"></a><span class='hs-comment'>-- <a href="http://www.math.tamu.edu/~sottile/research/pdf/SSym.pdf">http://www.math.tamu.edu/~sottile/research/pdf/SSym.pdf</a></span>
<a name="line-25"></a>
<a name="line-26"></a><span class='hs-comment'>-- Structure of the Loday-Ronco Hopf algebra of trees</span>
<a name="line-27"></a><span class='hs-comment'>-- Marcelo Aguiar and Frank Sottile</span>
<a name="line-28"></a><span class='hs-comment'>-- <a href="http://www.math.tamu.edu/~sottile/research/pdf/Loday.pdf">http://www.math.tamu.edu/~sottile/research/pdf/Loday.pdf</a></span>
<a name="line-29"></a>
<a name="line-30"></a><span class='hs-comment'>-- Hopf Structures on the Multiplihedra</span>
<a name="line-31"></a><span class='hs-comment'>-- Stefan Forcey, Aaron Lauve and Frank Sottile</span>
<a name="line-32"></a><span class='hs-comment'>-- <a href="http://www.math.tamu.edu/~sottile/research/pdf/MSym.pdf">http://www.math.tamu.edu/~sottile/research/pdf/MSym.pdf</a></span>
<a name="line-33"></a>
<a name="line-34"></a><span class='hs-comment'>-- Lie Algebras and Hopf Algebras</span>
<a name="line-35"></a><span class='hs-comment'>-- Michiel Hazewinkel, Nadiya Gubareni, V.V.Kirichenko</span>
<a name="line-36"></a>
<a name="line-37"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span> <span class='hs-varid'>hiding</span> <span class='hs-layout'>(</span> <span class='hs-layout'>(</span><span class='hs-varop'>*&gt;</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-38"></a>
<a name="line-39"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>L</span>
<a name="line-40"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Maybe</span> <span class='hs-layout'>(</span><span class='hs-varid'>fromJust</span><span class='hs-layout'>)</span>
<a name="line-41"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Set</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>S</span>
<a name="line-42"></a>
<a name="line-43"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Core</span><span class='hs-varop'>.</span><span class='hs-conid'>Field</span>
<a name="line-44"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Core</span><span class='hs-varop'>.</span><span class='hs-conid'>Utils</span>
<a name="line-45"></a>
<a name="line-46"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Algebras</span><span class='hs-varop'>.</span><span class='hs-conid'>VectorSpace</span> <span class='hs-varid'>hiding</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-layout'>)</span>
<a name="line-47"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Algebras</span><span class='hs-varop'>.</span><span class='hs-conid'>TensorProduct</span>
<a name="line-48"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Algebras</span><span class='hs-varop'>.</span><span class='hs-conid'>Structures</span>
<a name="line-49"></a>
<a name="line-50"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>Combinatorics</span><span class='hs-varop'>.</span><span class='hs-conid'>Poset</span>
<a name="line-51"></a>
<a name="line-52"></a><span class='hs-comment'>-- import Math.Algebra.Group.PermutationGroup</span>
<a name="line-53"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Math</span><span class='hs-varop'>.</span><span class='hs-conid'>CommutativeAlgebra</span><span class='hs-varop'>.</span><span class='hs-conid'>Polynomial</span>
<a name="line-54"></a>
<a name="line-55"></a>
<a name="line-56"></a><span class='hs-comment'>-- SHUFFLE ALGEBRA</span>
<a name="line-57"></a><span class='hs-comment'>-- This is just the tensor algebra, but with shuffle product (and deconcatenation coproduct)</span>
<a name="line-58"></a>
<a name="line-59"></a><a name="Shuffle"></a><span class='hs-comment'>-- |A basis for the shuffle algebra. As a vector space, the shuffle algebra is identical to the tensor algebra.</span>
<a name="line-60"></a><a name="Shuffle"></a><span class='hs-comment'>-- However, we consider a different algebra structure, based on the shuffle product. Together with the</span>
<a name="line-61"></a><a name="Shuffle"></a><span class='hs-comment'>-- deconcatenation coproduct, this leads to a Hopf algebra structure.</span>
<a name="line-62"></a><a name="Shuffle"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Sh</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span><span class='hs-conid'>Ord</span><span class='hs-layout'>,</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-63"></a>
<a name="line-64"></a><a name="sh"></a><span class='hs-comment'>-- |Construct a basis element of the shuffle algebra</span>
<a name="line-65"></a><span class='hs-definition'>sh</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-layout'>(</span><span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-66"></a><span class='hs-definition'>sh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Sh</span>
<a name="line-67"></a>
<a name="line-68"></a><a name="shuffles"></a><span class='hs-definition'>shuffles</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>shuffles</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>shuffles</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-69"></a><span class='hs-definition'>shuffles</span> <span class='hs-varid'>xs</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-70"></a><span class='hs-definition'>shuffles</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-71"></a>
<a name="line-72"></a><a name="instance%20Algebra%20k%20(Shuffle%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-73"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-74"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-75"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-conid'>Sh</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>shuffles</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-76"></a>
<a name="line-77"></a><a name="deconcatenations"></a><span class='hs-definition'>deconcatenations</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-varid'>inits</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>tails</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-78"></a>
<a name="line-79"></a><a name="instance%20Coalgebra%20k%20(Shuffle%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-80"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-81"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-82"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>us</span><span class='hs-layout'>,</span> <span class='hs-conid'>Sh</span> <span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>us</span><span class='hs-layout'>,</span> <span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>deconcatenations</span> <span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-83"></a>
<a name="line-84"></a><a name="instance%20Bialgebra%20k%20(Shuffle%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-85"></a>
<a name="line-86"></a><a name="instance%20HopfAlgebra%20k%20(Shuffle%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Shuffle</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-87"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varop'>^</span><span class='hs-varid'>length</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Sh</span> <span class='hs-layout'>(</span><span class='hs-varid'>reverse</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-88"></a>
<a name="line-89"></a>
<a name="line-90"></a><span class='hs-comment'>-- SSYM: PERMUTATIONS</span>
<a name="line-91"></a><span class='hs-comment'>-- (This is permutations considered as combinatorial objects rather than as algebraic objects)</span>
<a name="line-92"></a>
<a name="line-93"></a><span class='hs-comment'>-- Permutations with shifted shuffle product and flattened deconcatenation coproduct</span>
<a name="line-94"></a><span class='hs-comment'>-- This is the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.</span>
<a name="line-95"></a><span class='hs-comment'>-- It is neither commutative nor co-commutative</span>
<a name="line-96"></a>
<a name="line-97"></a><span class='hs-comment'>-- ssymF xs is the fundamental basis F_xs (Aguiar and Sottile)</span>
<a name="line-98"></a>
<a name="line-99"></a><a name="SSymF"></a><span class='hs-comment'>-- |The fundamental basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.</span>
<a name="line-100"></a><a name="SSymF"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-101"></a>
<a name="line-102"></a><a name="instance%20Ord%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-103"></a>    <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-104"></a>
<a name="line-105"></a><a name="instance%20Show%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-106"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"F "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>xs</span>
<a name="line-107"></a>
<a name="line-108"></a><a name="ssymF"></a><span class='hs-comment'>-- |Construct a fundamental basis element in SSym.</span>
<a name="line-109"></a><span class='hs-comment'>-- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].</span>
<a name="line-110"></a><span class='hs-definition'>ssymF</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>SSymF</span>
<a name="line-111"></a><span class='hs-definition'>ssymF</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>==</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-112"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"Not a permutation of [1..n]"</span>
<a name="line-113"></a>         <span class='hs-keyword'>where</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span>
<a name="line-114"></a>
<a name="line-115"></a><a name="shiftedConcat"></a><span class='hs-comment'>-- so this is a candidate mult. It is associative and SSymF [] is obviously a left and right identity</span>
<a name="line-116"></a><span class='hs-comment'>-- (need quickcheck properties to prove that)</span>
<a name="line-117"></a><span class='hs-definition'>shiftedConcat</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>in</span> <span class='hs-conid'>SSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-118"></a>
<a name="line-119"></a><a name="prop_Associative"></a><span class='hs-definition'>prop_Associative</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>,</span><span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>y</span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-varop'>==</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-varid'>z</span>
<a name="line-120"></a>
<a name="line-121"></a><span class='hs-comment'>-- &gt; quickCheck (prop_Associative shiftedConcat)</span>
<a name="line-122"></a><span class='hs-comment'>-- +++ OK, passed 100 tests.</span>
<a name="line-123"></a>
<a name="line-124"></a>
<a name="line-125"></a><a name="instance%20Algebra%20k%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-126"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-127"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-128"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-129"></a>            <span class='hs-keyword'>let</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span>
<a name="line-130"></a>            <span class='hs-keyword'>in</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>shuffles</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-131"></a>
<a name="line-132"></a>
<a name="line-133"></a><a name="flatten"></a><span class='hs-comment'>-- standard permutation, also called flattening, eg [6,2,5] -&gt; [3,1,2]</span>
<a name="line-134"></a><span class='hs-definition'>flatten</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>mapping</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span>
<a name="line-135"></a>        <span class='hs-keyword'>in</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-keyword'>let</span> <span class='hs-conid'>Just</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lookup</span> <span class='hs-varid'>x</span> <span class='hs-varid'>mapping</span><span class='hs-keyglyph'>]</span> 
<a name="line-136"></a>
<a name="line-137"></a><a name="instance%20Coalgebra%20k%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-138"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-139"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span>
<a name="line-140"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>st</span> <span class='hs-varid'>us</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>SSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>st</span> <span class='hs-varid'>vs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>us</span><span class='hs-layout'>,</span> <span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>deconcatenations</span> <span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-141"></a>              <span class='hs-varid'>st</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>flatten</span>
<a name="line-142"></a>
<a name="line-143"></a><a name="instance%20Bialgebra%20k%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-144"></a>
<a name="line-145"></a><a name="instance%20HopfAlgebra%20k%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-146"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-147"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-148"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-varid'>x</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>negatev</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>id</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>antipode</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>removeTerm</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>comult</span> <span class='hs-varop'>.</span> <span class='hs-varid'>return</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span>
<a name="line-149"></a>        <span class='hs-comment'>-- This expression for antipode is derived from mult . (id `tf` antipode) . comult == unit . counit</span>
<a name="line-150"></a>        <span class='hs-comment'>-- It's possible because this is a graded, connected Hopf algebra. (connected means the counit is projection onto the grade 0 part)</span>
<a name="line-151"></a><span class='hs-comment'>-- It would be nicer to have an explicit expression for antipode.</span>
<a name="line-152"></a><span class='hs-comment'>{-
<a name="line-153"></a>instance (Eq k, Num k) =&gt; HopfAlgebra k SSymF where
<a name="line-154"></a>    antipode = linear antipode'
<a name="line-155"></a>        where antipode' (SSymF v) = sumv [lambda v w *&gt; return (SSymF w) | w &lt;- L.permutations v]
<a name="line-156"></a>              lambda v w = length [s | s &lt;- powerset [1..n-1],  odd (length s), descentSet (w^-1 * v_s) `isSubset` s]
<a name="line-157"></a>                         - length [s | s &lt;- powerset [1..n-1],  even (length s), descentSet (w^-1 * v_s) `isSubset` s]
<a name="line-158"></a>-}</span>
<a name="line-159"></a>
<a name="line-160"></a><a name="instance%20HasInverses%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>HasInverses</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-161"></a>    <span class='hs-varid'>inverse</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SSymF</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-varid'>snd</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varop'>$</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-layout'>,</span><span class='hs-varid'>t</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>-&gt;</span><span class='hs-layout'>(</span><span class='hs-varid'>t</span><span class='hs-layout'>,</span><span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zip</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>xs</span>
<a name="line-162"></a>
<a name="line-163"></a><a name="instance%20HasPairing%20k%20SSymF%20SSymF"></a><span class='hs-comment'>-- Hazewinkel p267</span>
<a name="line-164"></a><a name="instance%20HasPairing%20k%20SSymF%20SSymF"></a><span class='hs-comment'>-- |A pairing showing that SSym is self-adjoint</span>
<a name="line-165"></a><a name="instance%20HasPairing%20k%20SSymF%20SSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HasPairing</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyword'>where</span>
<a name="line-166"></a>    <span class='hs-varid'>pairing</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>pairing'</span> <span class='hs-keyword'>where</span>
<a name="line-167"></a>        <span class='hs-varid'>pairing'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>delta</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>inverse</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span>
<a name="line-168"></a><span class='hs-comment'>-- Not entirely clear to me why this works</span>
<a name="line-169"></a><span class='hs-comment'>-- The pairing is *not* positive definite (Hazewinkel p267)</span>
<a name="line-170"></a><span class='hs-comment'>-- eg (\x -&gt; pairing' x x &gt;= 0) (ssymF [1,3,2] + ssymF [2,3,1] - ssymF [3,1,2]) == False</span>
<a name="line-171"></a>
<a name="line-172"></a>
<a name="line-173"></a><a name="SSymM"></a><span class='hs-comment'>-- |An alternative \"monomial\" basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.</span>
<a name="line-174"></a><a name="SSymM"></a><span class='hs-comment'>-- This basis is related to the fundamental basis by Mobius inversion in the poset of permutations with the weak order.</span>
<a name="line-175"></a><a name="SSymM"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-176"></a>
<a name="line-177"></a><a name="instance%20Ord%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span>
<a name="line-178"></a>    <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-179"></a>
<a name="line-180"></a><a name="instance%20Show%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span>
<a name="line-181"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"M "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>xs</span>
<a name="line-182"></a>
<a name="line-183"></a><a name="ssymM"></a><span class='hs-comment'>-- |Construct a monomial basis element in SSym.</span>
<a name="line-184"></a><span class='hs-comment'>-- The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].</span>
<a name="line-185"></a><span class='hs-definition'>ssymM</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>SSymM</span>
<a name="line-186"></a><span class='hs-definition'>ssymM</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>sort</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>==</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-187"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"Not a permutation of [1..n]"</span>
<a name="line-188"></a>         <span class='hs-keyword'>where</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span>
<a name="line-189"></a>
<a name="line-190"></a><a name="inversions"></a><span class='hs-definition'>inversions</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>ixs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zip</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>xs</span>
<a name="line-191"></a>                <span class='hs-keyword'>in</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>j</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-layout'>,</span><span class='hs-varid'>xi</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span><span class='hs-layout'>(</span><span class='hs-varid'>j</span><span class='hs-layout'>,</span><span class='hs-varid'>xj</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>pairs</span> <span class='hs-varid'>ixs</span><span class='hs-layout'>,</span> <span class='hs-varid'>xi</span> <span class='hs-varop'>&gt;</span> <span class='hs-varid'>xj</span><span class='hs-keyglyph'>]</span>
<a name="line-192"></a>
<a name="line-193"></a><a name="weakOrder"></a><span class='hs-comment'>-- should really check that xs and ys have the same length, and perhaps insist also on same type</span>
<a name="line-194"></a><span class='hs-definition'>weakOrder</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>inversions</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>`isSubsetAsc`</span> <span class='hs-varid'>inversions</span> <span class='hs-varid'>ys</span>
<a name="line-195"></a>
<a name="line-196"></a><a name="mu"></a><span class='hs-definition'>mu</span> <span class='hs-layout'>(</span><span class='hs-varid'>set</span><span class='hs-layout'>,</span><span class='hs-varid'>po</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mu'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>where</span>
<a name="line-197"></a>    <span class='hs-varid'>mu'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>x</span> <span class='hs-varop'>==</span> <span class='hs-varid'>y</span>    <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span>
<a name="line-198"></a>            <span class='hs-keyglyph'>|</span> <span class='hs-varid'>po</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>    <span class='hs-keyglyph'>=</span> <span class='hs-varid'>negate</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sum</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>mu'</span> <span class='hs-varid'>x</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>set</span><span class='hs-layout'>,</span> <span class='hs-varid'>po</span> <span class='hs-varid'>x</span> <span class='hs-varid'>z</span><span class='hs-layout'>,</span> <span class='hs-varid'>po</span> <span class='hs-varid'>z</span> <span class='hs-varid'>y</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-varop'>/=</span> <span class='hs-varid'>y</span><span class='hs-keyglyph'>]</span>
<a name="line-199"></a>            <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-200"></a>
<a name="line-201"></a><a name="ssymMtoF"></a><span class='hs-comment'>-- |Convert an element of SSym represented in the monomial basis to the fundamental basis</span>
<a name="line-202"></a><span class='hs-definition'>ssymMtoF</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span>
<a name="line-203"></a><span class='hs-definition'>ssymMtoF</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>ssymMtoF'</span> <span class='hs-keyword'>where</span>
<a name="line-204"></a>    <span class='hs-varid'>ssymMtoF'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>mu</span> <span class='hs-layout'>(</span><span class='hs-varid'>set</span><span class='hs-layout'>,</span><span class='hs-varid'>po</span><span class='hs-layout'>)</span> <span class='hs-varid'>u</span> <span class='hs-varid'>v</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>set</span><span class='hs-layout'>,</span> <span class='hs-varid'>po</span> <span class='hs-varid'>u</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span>
<a name="line-205"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>set</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>permutations</span> <span class='hs-varid'>u</span>
<a name="line-206"></a>              <span class='hs-varid'>po</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>weakOrder</span>
<a name="line-207"></a>
<a name="line-208"></a><a name="ssymFtoM"></a><span class='hs-comment'>-- |Convert an element of SSym represented in the fundamental basis to the monomial basis</span>
<a name="line-209"></a><span class='hs-definition'>ssymFtoM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span>
<a name="line-210"></a><span class='hs-definition'>ssymFtoM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>ssymFtoM'</span> <span class='hs-keyword'>where</span>
<a name="line-211"></a>    <span class='hs-varid'>ssymFtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>set</span><span class='hs-layout'>,</span> <span class='hs-varid'>po</span> <span class='hs-varid'>u</span> <span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span>
<a name="line-212"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>set</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>permutations</span> <span class='hs-varid'>u</span>
<a name="line-213"></a>              <span class='hs-varid'>po</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>weakOrder</span>
<a name="line-214"></a>
<a name="line-215"></a><span class='hs-comment'>-- (p,q)-shuffles: permutations of [1..p+q] having at most one descent, at position p</span>
<a name="line-216"></a><span class='hs-comment'>-- denoted S^{(p,q)} in Aguiar&amp;Sottile</span>
<a name="line-217"></a><span class='hs-comment'>-- (Grassmannian permutations?)</span>
<a name="line-218"></a><span class='hs-comment'>-- pqShuffles p q = [u++v | u &lt;- combinationsOf p [1..n], let v = [1..n] `diffAsc` u] where n = p+q</span>
<a name="line-219"></a>
<a name="line-220"></a><span class='hs-comment'>-- The inverse of a (p,q)-shuffle.</span>
<a name="line-221"></a><span class='hs-comment'>-- The special form of (p,q)-shuffles makes an O(n) algorithm possible</span>
<a name="line-222"></a><span class='hs-comment'>-- pqInverse :: Int -&gt; Int -&gt; [Int] -&gt; [Int]</span>
<a name="line-223"></a><span class='hs-comment'>{-
<a name="line-224"></a>-- incorrect
<a name="line-225"></a>pqInverse p q xs = pqInverse' [1..p] [p+1..p+q] xs
<a name="line-226"></a>    where pqInverse' (l:ls) (r:rs) (x:xs) =
<a name="line-227"></a>              if x &lt;= p then l : pqInverse' ls (r:rs) xs else r : pqInverse' (l:ls) rs xs
<a name="line-228"></a>          pqInverse' ls rs _ = ls ++ rs -- one of them is null
<a name="line-229"></a>-}</span>
<a name="line-230"></a><span class='hs-comment'>-- pqInverseShuffles p q = shuffles [1..p] [p+1..p+q]</span>
<a name="line-231"></a>
<a name="line-232"></a>
<a name="line-233"></a><a name="instance%20Algebra%20k%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span>
<a name="line-234"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-235"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ssymFtoM</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>ssymMtoF</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>ssymMtoF</span><span class='hs-layout'>)</span>
<a name="line-236"></a>
<a name="line-237"></a><span class='hs-comment'>{-
<a name="line-238"></a>mult2 = linear mult'
<a name="line-239"></a>    where mult' (SSymM u, SSymM v) = sumv [alpha u v w *&gt; return (SSymM w) | w &lt;- L.permutations [1..p+q] ]
<a name="line-240"></a>                                     where p = length u; q = length v
<a name="line-241"></a>
<a name="line-242"></a>alpha u v w = length [z | z &lt;- pqInverseShuffles p q, let uv = shiftedConcat u v,
<a name="line-243"></a>                          uv * z `weakOrder` w, u and v are maximal, ie no transposition of adjacents in either also works]
<a name="line-244"></a>    where p = length u
<a name="line-245"></a>          q = length v
<a name="line-246"></a>-- so we need to define (*) for permutations in row form
<a name="line-247"></a>-}</span>
<a name="line-248"></a>
<a name="line-249"></a><a name="instance%20Coalgebra%20k%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span>
<a name="line-250"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-251"></a>    <span class='hs-comment'>-- comult = (ssymFtoM `tf` ssymFtoM) . comult . ssymMtoF</span>
<a name="line-252"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span>
<a name="line-253"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymM</span> <span class='hs-layout'>(</span><span class='hs-varid'>flatten</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>SSymM</span> <span class='hs-layout'>(</span><span class='hs-varid'>flatten</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-254"></a>                                        <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span><span class='hs-layout'>,</span><span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>deconcatenations</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span>
<a name="line-255"></a>                                          <span class='hs-varid'>minimum</span> <span class='hs-layout'>(</span><span class='hs-varid'>infinity</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-varop'>&gt;</span> <span class='hs-varid'>maximum</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-conop'>:</span><span class='hs-varid'>zs</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ie deconcatenations at a global descent</span>
<a name="line-256"></a>              <span class='hs-varid'>infinity</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>maxBound</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span>
<a name="line-257"></a>
<a name="line-258"></a><a name="instance%20Bialgebra%20k%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-259"></a>
<a name="line-260"></a><a name="instance%20HopfAlgebra%20k%20SSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymM</span> <span class='hs-keyword'>where</span>
<a name="line-261"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ssymFtoM</span> <span class='hs-varop'>.</span> <span class='hs-varid'>antipode</span> <span class='hs-varop'>.</span> <span class='hs-varid'>ssymMtoF</span>
<a name="line-262"></a>
<a name="line-263"></a>
<a name="line-264"></a><a name="instance%20Algebra%20k%20(Dual%20SSymF)"></a><span class='hs-comment'>-- Hazewinkel p265</span>
<a name="line-265"></a><a name="instance%20Algebra%20k%20(Dual%20SSymF)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-266"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-267"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-268"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-269"></a>            <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>return</span> <span class='hs-varop'>.</span> <span class='hs-conid'>Dual</span> <span class='hs-varop'>.</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs''</span> <span class='hs-varop'>++</span> <span class='hs-varid'>ys''</span><span class='hs-layout'>)</span>
<a name="line-270"></a>                 <span class='hs-keyglyph'>|</span> <span class='hs-varid'>xs'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>combinationsOf</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>r</span><span class='hs-varop'>+</span><span class='hs-varid'>s</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>ys'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>diffAsc</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>r</span><span class='hs-varop'>+</span><span class='hs-varid'>s</span><span class='hs-keyglyph'>]</span> <span class='hs-varid'>xs'</span><span class='hs-layout'>,</span>
<a name="line-271"></a>                   <span class='hs-varid'>xs''</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>permutations</span> <span class='hs-varid'>xs'</span><span class='hs-layout'>,</span> <span class='hs-varid'>flatten</span> <span class='hs-varid'>xs''</span> <span class='hs-varop'>==</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span>
<a name="line-272"></a>                   <span class='hs-varid'>ys''</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>permutations</span> <span class='hs-varid'>ys'</span><span class='hs-layout'>,</span> <span class='hs-varid'>flatten</span> <span class='hs-varid'>ys''</span> <span class='hs-varop'>==</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>]</span>
<a name="line-273"></a>            <span class='hs-keyword'>where</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span><span class='hs-layout'>;</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>ys</span>
<a name="line-274"></a><span class='hs-comment'>-- In other words, mult x y is the sum of those z whose comult (in SSymF) has an (x,y) term</span>
<a name="line-275"></a><span class='hs-comment'>-- So the matrix for mult is the transpose of the matrix for comult in SSymF</span>
<a name="line-276"></a>
<a name="line-277"></a><a name="instance%20Coalgebra%20k%20(Dual%20SSymF)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-278"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-279"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-280"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-281"></a>            <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>flatten</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>ys</span><span class='hs-layout'>,</span><span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>partition</span> <span class='hs-layout'>(</span><span class='hs-varop'>&lt;=</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>]</span>
<a name="line-282"></a>            <span class='hs-keyword'>where</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>xs</span>
<a name="line-283"></a><span class='hs-comment'>-- In other words, comult x is the sum of those (y,z) whose mult (in SSymF) has a z term</span>
<a name="line-284"></a><span class='hs-comment'>-- So the matrix for comult is the transpose of the matrix for mult in SSymF</span>
<a name="line-285"></a>
<a name="line-286"></a><a name="instance%20Bialgebra%20k%20(Dual%20SSymF)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-287"></a>
<a name="line-288"></a><a name="instance%20HopfAlgebra%20k%20(Dual%20SSymF)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-289"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-290"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-291"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-varid'>x</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span>
<a name="line-292"></a>            <span class='hs-layout'>(</span><span class='hs-varid'>negatev</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>id</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>antipode</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>removeTerm</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>comult</span> <span class='hs-varop'>.</span> <span class='hs-varid'>return</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span>
<a name="line-293"></a>
<a name="line-294"></a><a name="instance%20HasPairing%20k%20SSymF%20(Dual%20SSymF)"></a><span class='hs-comment'>-- This pairing is positive definite (Hazewinkel p267)</span>
<a name="line-295"></a><a name="instance%20HasPairing%20k%20SSymF%20(Dual%20SSymF)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HasPairing</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-296"></a>    <span class='hs-varid'>pairing</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>pairing'</span> <span class='hs-keyword'>where</span>
<a name="line-297"></a>        <span class='hs-varid'>pairing'</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-conid'>Dual</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>delta</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span>
<a name="line-298"></a>
<a name="line-299"></a><a name="ssymFtoDual"></a><span class='hs-comment'>-- |The isomorphism from SSym to its dual that takes a permutation in the fundamental basis to its inverse in the dual basis</span>
<a name="line-300"></a><span class='hs-definition'>ssymFtoDual</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-conid'>SSymF</span><span class='hs-layout'>)</span>
<a name="line-301"></a><span class='hs-definition'>ssymFtoDual</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-conid'>Dual</span> <span class='hs-varop'>.</span> <span class='hs-varid'>inverse</span><span class='hs-layout'>)</span>
<a name="line-302"></a><span class='hs-comment'>-- This is theta on Hazewinkel p266 (though later he also uses theta for the inverse of this map)</span>
<a name="line-303"></a>
<a name="line-304"></a>
<a name="line-305"></a><span class='hs-comment'>-- YSYM: PLANAR BINARY TREES</span>
<a name="line-306"></a><span class='hs-comment'>-- These are really rooted planar binary trees.</span>
<a name="line-307"></a><span class='hs-comment'>-- It's because they're planar that we can distinguish left and right child branches.</span>
<a name="line-308"></a><span class='hs-comment'>-- (Non-planar would be if we considered trees where left and right children are swapped relative to one another as the same tree)</span>
<a name="line-309"></a><span class='hs-comment'>-- It is neither commutative nor co-commutative</span>
<a name="line-310"></a>
<a name="line-311"></a><a name="PBT"></a><span class='hs-comment'>-- |A type for (rooted) planar binary trees. The basis elements of the Loday-Ronco Hopf algebra are indexed by these.</span>
<a name="line-312"></a><a name="PBT"></a><span class='hs-comment'>--</span>
<a name="line-313"></a><a name="PBT"></a><span class='hs-comment'>-- Although the trees are labelled, we're really only interested in the shapes of the trees, and hence in the type PBT ().</span>
<a name="line-314"></a><a name="PBT"></a><span class='hs-comment'>-- The Algebra, Coalgebra and HopfAlgebra instances all ignore the labels.</span>
<a name="line-315"></a><a name="PBT"></a><span class='hs-comment'>-- However, it is convenient to allow labels, as they can be useful for seeing what is going on, and they also make it possible</span>
<a name="line-316"></a><a name="PBT"></a><span class='hs-comment'>-- to define various ways to create trees from lists of labels.</span>
<a name="line-317"></a><a name="PBT"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-layout'>(</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>E</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Show</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span><span class='hs-layout'>)</span>
<a name="line-318"></a>
<a name="line-319"></a><a name="instance%20Ord%20(PBT%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Ord</span> <span class='hs-layout'>(</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-320"></a>    <span class='hs-varid'>compare</span> <span class='hs-varid'>u</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>shapeSignature</span> <span class='hs-varid'>u</span><span class='hs-layout'>,</span> <span class='hs-varid'>prefix</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>shapeSignature</span> <span class='hs-varid'>v</span><span class='hs-layout'>,</span> <span class='hs-varid'>prefix</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-321"></a>
<a name="line-322"></a><a name="YSymF"></a><span class='hs-comment'>-- |The fundamental basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym.</span>
<a name="line-323"></a><a name="YSymF"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>YSymF</span> <span class='hs-layout'>(</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span><span class='hs-layout'>)</span>
<a name="line-324"></a>
<a name="line-325"></a><a name="instance%20Show%20(YSymF%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Show</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-326"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"F("</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>t</span> <span class='hs-varop'>++</span> <span class='hs-str'>")"</span>
<a name="line-327"></a>
<a name="line-328"></a><a name="ysymF"></a><span class='hs-comment'>-- |Construct the element of YSym in the fundamental basis indexed by the given tree</span>
<a name="line-329"></a><span class='hs-definition'>ysymF</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-330"></a><span class='hs-definition'>ysymF</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-331"></a>
<a name="line-332"></a><a name="nodecount"></a><span class='hs-comment'>{-
<a name="line-333"></a>depth (T l x r) = 1 + max (depth l) (depth r)
<a name="line-334"></a>depth E = 0
<a name="line-335"></a>-}</span>
<a name="line-336"></a><span class='hs-definition'>nodecount</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-varid'>nodecount</span> <span class='hs-varid'>l</span> <span class='hs-varop'>+</span> <span class='hs-varid'>nodecount</span> <span class='hs-varid'>r</span>
<a name="line-337"></a><span class='hs-definition'>nodecount</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-338"></a>
<a name="line-339"></a><a name="leafcount"></a><span class='hs-comment'>-- in fact leafcount t = 1 + nodecount t (easiest to see with a picture)</span>
<a name="line-340"></a><span class='hs-definition'>leafcount</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leafcount</span> <span class='hs-varid'>l</span> <span class='hs-varop'>+</span> <span class='hs-varid'>leafcount</span> <span class='hs-varid'>r</span>
<a name="line-341"></a><span class='hs-definition'>leafcount</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span>
<a name="line-342"></a>
<a name="line-343"></a><a name="prefix"></a><span class='hs-definition'>prefix</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-344"></a><span class='hs-definition'>prefix</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-varid'>prefix</span> <span class='hs-varid'>l</span> <span class='hs-varop'>++</span> <span class='hs-varid'>prefix</span> <span class='hs-varid'>r</span>
<a name="line-345"></a>
<a name="line-346"></a><a name="shapeSignature"></a><span class='hs-comment'>-- The shape signature uniquely identifies the shape of a tree.</span>
<a name="line-347"></a><span class='hs-comment'>-- Trees with distinct shapes have distinct signatures.</span>
<a name="line-348"></a><span class='hs-comment'>-- In addition, if sorting on shapeSignature, smaller trees sort before larger trees,</span>
<a name="line-349"></a><span class='hs-comment'>-- and leftward leaning trees sort before rightward leaning trees</span>
<a name="line-350"></a><span class='hs-definition'>shapeSignature</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>shapeSignature'</span> <span class='hs-layout'>(</span><span class='hs-varid'>nodeCountTree</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-351"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>shapeSignature'</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- not [], otherwise we can't distinguish T (T E () E) () E from T E () (T E () E)</span>
<a name="line-352"></a>          <span class='hs-varid'>shapeSignature'</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-conop'>:</span> <span class='hs-varid'>shapeSignature'</span> <span class='hs-varid'>r</span> <span class='hs-varop'>++</span> <span class='hs-varid'>shapeSignature'</span> <span class='hs-varid'>l</span>
<a name="line-353"></a>
<a name="line-354"></a><a name="nodeCountTree"></a><span class='hs-definition'>nodeCountTree</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span>
<a name="line-355"></a><span class='hs-definition'>nodeCountTree</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>n</span> <span class='hs-varid'>r'</span>
<a name="line-356"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>l'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nodeCountTree</span> <span class='hs-varid'>l</span>
<a name="line-357"></a>          <span class='hs-varid'>r'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nodeCountTree</span> <span class='hs-varid'>r</span>
<a name="line-358"></a>          <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>l'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>lc</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>lc</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>r'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>rc</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>rc</span><span class='hs-layout'>)</span>
<a name="line-359"></a>
<a name="line-360"></a><a name="leafCountTree"></a><span class='hs-definition'>leafCountTree</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span>
<a name="line-361"></a><span class='hs-definition'>leafCountTree</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l'</span> <span class='hs-varid'>n</span> <span class='hs-varid'>r'</span>
<a name="line-362"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>l'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leafCountTree</span> <span class='hs-varid'>l</span>
<a name="line-363"></a>          <span class='hs-varid'>r'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leafCountTree</span> <span class='hs-varid'>r</span>
<a name="line-364"></a>          <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>l'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>1</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>lc</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>lc</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>r'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>1</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>rc</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>rc</span><span class='hs-layout'>)</span>
<a name="line-365"></a>
<a name="line-366"></a><a name="lrCountTree"></a><span class='hs-comment'>-- A tree that counts nodes in left and right subtrees</span>
<a name="line-367"></a><span class='hs-definition'>lrCountTree</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span>
<a name="line-368"></a><span class='hs-definition'>lrCountTree</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l'</span> <span class='hs-layout'>(</span><span class='hs-varid'>lc</span><span class='hs-layout'>,</span><span class='hs-varid'>rc</span><span class='hs-layout'>)</span> <span class='hs-varid'>r'</span>
<a name="line-369"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>l'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lrCountTree</span> <span class='hs-varid'>l</span>
<a name="line-370"></a>          <span class='hs-varid'>r'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>lrCountTree</span> <span class='hs-varid'>r</span>
<a name="line-371"></a>          <span class='hs-varid'>lc</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>l'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-varid'>llc</span><span class='hs-layout'>,</span><span class='hs-varid'>lrc</span><span class='hs-layout'>)</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-varid'>llc</span> <span class='hs-varop'>+</span> <span class='hs-varid'>lrc</span>
<a name="line-372"></a>          <span class='hs-varid'>rc</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>r'</span> <span class='hs-keyword'>of</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>0</span><span class='hs-layout'>;</span> <span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-varid'>rlc</span><span class='hs-layout'>,</span><span class='hs-varid'>rrc</span><span class='hs-layout'>)</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-num'>1</span> <span class='hs-varop'>+</span> <span class='hs-varid'>rlc</span> <span class='hs-varop'>+</span> <span class='hs-varid'>rrc</span>
<a name="line-373"></a>
<a name="line-374"></a><a name="shape"></a><span class='hs-definition'>shape</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>PBT</span> <span class='hs-conid'>()</span>
<a name="line-375"></a><span class='hs-definition'>shape</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-keyword'>_</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>()</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span>
<a name="line-376"></a>
<a name="line-377"></a><a name="numbered"></a><span class='hs-comment'>-- label the nodes of a tree in infix order while preserving its shape</span>
<a name="line-378"></a><span class='hs-definition'>numbered</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>numbered'</span> <span class='hs-num'>1</span> <span class='hs-varid'>t</span>
<a name="line-379"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>numbered'</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span>
<a name="line-380"></a>          <span class='hs-varid'>numbered'</span> <span class='hs-varid'>i</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nodecount</span> <span class='hs-varid'>l</span> <span class='hs-keyword'>in</span> <span class='hs-conid'>T</span> <span class='hs-layout'>(</span><span class='hs-varid'>numbered'</span> <span class='hs-varid'>i</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>numbered'</span> <span class='hs-layout'>(</span><span class='hs-varid'>i</span><span class='hs-varop'>+</span><span class='hs-varid'>k</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-381"></a><span class='hs-comment'>-- could also pair the numbers with the input labels</span>
<a name="line-382"></a>
<a name="line-383"></a>
<a name="line-384"></a><a name="splits"></a><span class='hs-definition'>splits</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-layout'>,</span><span class='hs-conid'>E</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-385"></a><span class='hs-definition'>splits</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span> <span class='hs-conid'>T</span> <span class='hs-varid'>v</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>splits</span> <span class='hs-varid'>l</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>u</span><span class='hs-layout'>,</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>splits</span> <span class='hs-varid'>r</span><span class='hs-keyglyph'>]</span>
<a name="line-386"></a>
<a name="line-387"></a><a name="instance%20Coalgebra%20k%20(YSymF%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-388"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span><span class='hs-layout'>;</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-389"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span>
<a name="line-390"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>u</span><span class='hs-layout'>,</span> <span class='hs-conid'>YSymF</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>splits</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span>
<a name="line-391"></a>              <span class='hs-comment'>-- using sumv rather than sum to avoid requiring Show a</span>
<a name="line-392"></a>    <span class='hs-comment'>-- so again this is a kind of deconcatenation coproduct</span>
<a name="line-393"></a>
<a name="line-394"></a><a name="multisplits"></a><span class='hs-definition'>multisplits</span> <span class='hs-num'>1</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-395"></a><span class='hs-definition'>multisplits</span> <span class='hs-num'>2</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>splits</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>]</span>
<a name="line-396"></a><span class='hs-definition'>multisplits</span> <span class='hs-varid'>n</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span> <span class='hs-varid'>u</span><span class='hs-conop'>:</span><span class='hs-varid'>ws</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>u</span><span class='hs-layout'>,</span><span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>splits</span> <span class='hs-varid'>t</span><span class='hs-layout'>,</span> <span class='hs-varid'>ws</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>multisplits</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>]</span>
<a name="line-397"></a>
<a name="line-398"></a><a name="graft"></a><span class='hs-definition'>graft</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>t</span>
<a name="line-399"></a><span class='hs-definition'>graft</span> <span class='hs-varid'>ts</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>ls</span><span class='hs-layout'>,</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>splitAt</span> <span class='hs-layout'>(</span><span class='hs-varid'>leafcount</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>ts</span>
<a name="line-400"></a>                     <span class='hs-keyword'>in</span> <span class='hs-conid'>T</span> <span class='hs-layout'>(</span><span class='hs-varid'>graft</span> <span class='hs-varid'>ls</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>graft</span> <span class='hs-varid'>rs</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-401"></a>
<a name="line-402"></a><a name="instance%20Algebra%20k%20(YSymF%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-403"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span>
<a name="line-404"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-405"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>,</span> <span class='hs-conid'>YSymF</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>graft</span> <span class='hs-varid'>ts</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>ts</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>multisplits</span> <span class='hs-layout'>(</span><span class='hs-varid'>leafcount</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span>
<a name="line-406"></a>        <span class='hs-comment'>-- using sumv rather than sum to avoid requiring Show a</span>
<a name="line-407"></a>
<a name="line-408"></a><a name="instance%20Bialgebra%20k%20(YSymF%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-409"></a>
<a name="line-410"></a><a name="instance%20HopfAlgebra%20k%20(YSymF%20a)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-411"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-412"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span>
<a name="line-413"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>negatev</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>id</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>antipode</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>removeTerm</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>E</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>comult</span> <span class='hs-varop'>.</span> <span class='hs-varid'>return</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span>
<a name="line-414"></a>
<a name="line-415"></a>
<a name="line-416"></a><a name="YSymM"></a><span class='hs-comment'>-- |An alternative \"monomial\" basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym.</span>
<a name="line-417"></a><a name="YSymM"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>YSymM</span> <span class='hs-layout'>(</span><span class='hs-conid'>PBT</span> <span class='hs-conid'>()</span><span class='hs-layout'>)</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span> <span class='hs-conid'>Ord</span><span class='hs-layout'>)</span>
<a name="line-418"></a>
<a name="line-419"></a><a name="instance%20Show%20YSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyword'>where</span>
<a name="line-420"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"M("</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>t</span> <span class='hs-varop'>++</span> <span class='hs-str'>")"</span>
<a name="line-421"></a>
<a name="line-422"></a><a name="ysymM"></a><span class='hs-comment'>-- |Construct the element of YSym in the monomial basis indexed by the given tree</span>
<a name="line-423"></a><span class='hs-definition'>ysymM</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PBT</span> <span class='hs-conid'>()</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>YSymM</span>
<a name="line-424"></a><span class='hs-definition'>ysymM</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-425"></a>
<a name="line-426"></a><a name="trees"></a><span class='hs-comment'>-- |List all trees with the given number of nodes</span>
<a name="line-427"></a><span class='hs-definition'>trees</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>PBT</span> <span class='hs-conid'>()</span><span class='hs-keyglyph'>]</span>
<a name="line-428"></a><span class='hs-definition'>trees</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>E</span><span class='hs-keyglyph'>]</span>
<a name="line-429"></a><span class='hs-definition'>trees</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-conid'>()</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-varid'>l</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>trees</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-comment'>-</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>trees</span> <span class='hs-varid'>i</span><span class='hs-keyglyph'>]</span>
<a name="line-430"></a>
<a name="line-431"></a><a name="tamariCovers"></a><span class='hs-comment'>-- |The covering relation for the Tamari partial order on binary trees</span>
<a name="line-432"></a><span class='hs-definition'>tamariCovers</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-433"></a><span class='hs-definition'>tamariCovers</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-434"></a><span class='hs-definition'>tamariCovers</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>u</span> <span class='hs-varid'>x</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span> <span class='hs-varid'>y</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>T</span> <span class='hs-varid'>t'</span> <span class='hs-varid'>y</span> <span class='hs-varid'>w</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>t'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>tamariCovers</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span>
<a name="line-435"></a>                                <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>T</span> <span class='hs-varid'>t</span> <span class='hs-varid'>y</span> <span class='hs-varid'>w'</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>w'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>tamariCovers</span> <span class='hs-varid'>w</span><span class='hs-keyglyph'>]</span>
<a name="line-436"></a>                                <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>T</span> <span class='hs-varid'>u</span> <span class='hs-varid'>y</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>v</span> <span class='hs-varid'>x</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-437"></a>                                <span class='hs-comment'>-- Note that this preserves the descending property, and hence the bijection with permutations</span>
<a name="line-438"></a>                                <span class='hs-comment'>-- If we were to swap x and y, we would preserve the binary search tree property instead (if our trees had it)</span>
<a name="line-439"></a><span class='hs-definition'>tamariCovers</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-conid'>E</span> <span class='hs-varid'>x</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>T</span> <span class='hs-conid'>E</span> <span class='hs-varid'>x</span> <span class='hs-varid'>u'</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>u'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>tamariCovers</span> <span class='hs-varid'>u</span><span class='hs-keyglyph'>]</span>  
<a name="line-440"></a>
<a name="line-441"></a><a name="tamariUpSet"></a><span class='hs-comment'>-- |The up-set of a binary tree in the Tamari partial order</span>
<a name="line-442"></a><span class='hs-definition'>tamariUpSet</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span>
<a name="line-443"></a><span class='hs-definition'>tamariUpSet</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>upSet'</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span>
<a name="line-444"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>upSet'</span> <span class='hs-varid'>interior</span> <span class='hs-varid'>boundary</span> <span class='hs-keyglyph'>=</span>
<a name="line-445"></a>              <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>boundary</span>
<a name="line-446"></a>              <span class='hs-keyword'>then</span> <span class='hs-varid'>interior</span>
<a name="line-447"></a>              <span class='hs-keyword'>else</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>interior'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>setUnionAsc</span> <span class='hs-varid'>interior</span> <span class='hs-varid'>boundary</span>
<a name="line-448"></a>                       <span class='hs-varid'>boundary'</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>toSet</span> <span class='hs-varop'>$</span> <span class='hs-varid'>concatMap</span> <span class='hs-varid'>tamariCovers</span> <span class='hs-varid'>boundary</span>
<a name="line-449"></a>                   <span class='hs-keyword'>in</span> <span class='hs-varid'>upSet'</span> <span class='hs-varid'>interior'</span> <span class='hs-varid'>boundary'</span>
<a name="line-450"></a>
<a name="line-451"></a><span class='hs-comment'>-- tamariOrder1 u v = v `elem` upSet u</span>
<a name="line-452"></a>
<a name="line-453"></a><a name="tamariOrder"></a><span class='hs-comment'>-- |The Tamari partial order on binary trees.</span>
<a name="line-454"></a><span class='hs-comment'>-- This is only defined between trees of the same size (number of nodes).</span>
<a name="line-455"></a><span class='hs-comment'>-- The result between trees of different sizes is undefined (we don't check).</span>
<a name="line-456"></a><span class='hs-definition'>tamariOrder</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>PBT</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Bool</span>
<a name="line-457"></a><span class='hs-definition'>tamariOrder</span> <span class='hs-varid'>u</span> <span class='hs-varid'>v</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>weakOrder</span> <span class='hs-layout'>(</span><span class='hs-varid'>minPerm</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>minPerm</span> <span class='hs-varid'>v</span><span class='hs-layout'>)</span>
<a name="line-458"></a><span class='hs-comment'>-- It should be possible to unpack this to be a statement purely about trees, but probably not worth it</span>
<a name="line-459"></a>
<a name="line-460"></a><a name="ysymMtoF"></a><span class='hs-comment'>-- |Convert an element of YSym represented in the monomial basis to the fundamental basis</span>
<a name="line-461"></a><span class='hs-definition'>ysymMtoF</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>()</span><span class='hs-layout'>)</span>
<a name="line-462"></a><span class='hs-definition'>ysymMtoF</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>ysymMtoF'</span> <span class='hs-keyword'>where</span>
<a name="line-463"></a>    <span class='hs-varid'>ysymMtoF'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>mu</span> <span class='hs-layout'>(</span><span class='hs-varid'>set</span><span class='hs-layout'>,</span><span class='hs-varid'>po</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span> <span class='hs-varid'>s</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>set</span><span class='hs-keyglyph'>]</span>
<a name="line-464"></a>        <span class='hs-keyword'>where</span> <span class='hs-varid'>po</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>tamariOrder</span>
<a name="line-465"></a>              <span class='hs-varid'>set</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>tamariUpSet</span> <span class='hs-varid'>t</span> <span class='hs-comment'>-- [s | s &lt;- trees (nodecount t), t `tamariOrder` s]</span>
<a name="line-466"></a>
<a name="line-467"></a><a name="ysymFtoM"></a><span class='hs-comment'>-- |Convert an element of YSym represented in the fundamental basis to the monomial basis</span>
<a name="line-468"></a><span class='hs-definition'>ysymFtoM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>()</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span>
<a name="line-469"></a><span class='hs-definition'>ysymFtoM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>ysymFtoM'</span> <span class='hs-keyword'>where</span>
<a name="line-470"></a>    <span class='hs-varid'>ysymFtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>tamariUpSet</span> <span class='hs-varid'>t</span><span class='hs-keyglyph'>]</span>
<a name="line-471"></a>                       <span class='hs-comment'>-- sumv [return (YSymM s) | s &lt;- trees (nodecount t), t `tamariOrder` s]</span>
<a name="line-472"></a>
<a name="line-473"></a>
<a name="line-474"></a><a name="instance%20Algebra%20k%20YSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyword'>where</span>
<a name="line-475"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span>
<a name="line-476"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ysymFtoM</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>ysymMtoF</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>ysymMtoF</span><span class='hs-layout'>)</span>
<a name="line-477"></a>
<a name="line-478"></a><a name="instance%20Coalgebra%20k%20YSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyword'>where</span>
<a name="line-479"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span><span class='hs-layout'>;</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span>
<a name="line-480"></a>    <span class='hs-comment'>-- comult = (ysymFtoM `tf` ysymFtoM) . comult . ysymMtoF</span>
<a name="line-481"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-482"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>r</span><span class='hs-layout'>,</span> <span class='hs-conid'>YSymM</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span><span class='hs-layout'>,</span><span class='hs-varid'>ss</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>deconcatenations</span> <span class='hs-layout'>(</span><span class='hs-varid'>underDecomposition</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-483"></a>                                                              <span class='hs-keyword'>let</span> <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldl</span> <span class='hs-varid'>under</span> <span class='hs-conid'>E</span> <span class='hs-varid'>rs</span><span class='hs-layout'>,</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldl</span> <span class='hs-varid'>under</span> <span class='hs-conid'>E</span> <span class='hs-varid'>ss</span><span class='hs-keyglyph'>]</span>
<a name="line-484"></a>
<a name="line-485"></a>
<a name="line-486"></a><a name="instance%20Bialgebra%20k%20YSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-487"></a>
<a name="line-488"></a><a name="instance%20HopfAlgebra%20k%20YSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>YSymM</span> <span class='hs-keyword'>where</span>
<a name="line-489"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>ysymFtoM</span> <span class='hs-varop'>.</span> <span class='hs-varid'>antipode</span> <span class='hs-varop'>.</span> <span class='hs-varid'>ysymMtoF</span> 
<a name="line-490"></a>
<a name="line-491"></a>
<a name="line-492"></a><span class='hs-comment'>-- QSYM: QUASI-SYMMETRIC FUNCTIONS</span>
<a name="line-493"></a><span class='hs-comment'>-- The following is the Hopf algebra QSym of quasi-symmetric functions</span>
<a name="line-494"></a><span class='hs-comment'>-- using the monomial and fundamental bases (indexed by compositions)</span>
<a name="line-495"></a>
<a name="line-496"></a><a name="compositions"></a><span class='hs-comment'>-- compositions in ascending order</span>
<a name="line-497"></a><span class='hs-comment'>-- might be better to use bfs to get length order</span>
<a name="line-498"></a><span class='hs-comment'>-- |List the compositions of an integer n. For example, the compositions of 4 are [[1,1,1,1],[1,1,2],[1,2,1],[1,3],[2,1,1],[2,2],[3,1],[4]]</span>
<a name="line-499"></a><span class='hs-definition'>compositions</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span>
<a name="line-500"></a><span class='hs-definition'>compositions</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>[]</span><span class='hs-keyglyph'>]</span>
<a name="line-501"></a><span class='hs-definition'>compositions</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>i</span><span class='hs-conop'>:</span><span class='hs-varid'>is</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span> <span class='hs-varid'>is</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>compositions</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-502"></a>
<a name="line-503"></a><span class='hs-comment'>-- can retrieve subsets of [1..n-1] from compositions n as follows</span>
<a name="line-504"></a><span class='hs-comment'>-- &gt; map (tail . scanl (+) 0) (map init $ compositions 4)</span>
<a name="line-505"></a><span class='hs-comment'>-- [[],[3],[2],[2,3],[1],[1,3],[1,2],[1,2,3]]</span>
<a name="line-506"></a>
<a name="line-507"></a><a name="quasiShuffles"></a><span class='hs-comment'>-- quasi shuffles of two compositions</span>
<a name="line-508"></a><span class='hs-definition'>quasiShuffles</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span><span class='hs-keyglyph'>]</span>
<a name="line-509"></a><span class='hs-definition'>quasiShuffles</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>quasiShuffles</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span>
<a name="line-510"></a>                              <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-varop'>+</span><span class='hs-varid'>y</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>quasiShuffles</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span>
<a name="line-511"></a>                              <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>quasiShuffles</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-512"></a><span class='hs-definition'>quasiShuffles</span> <span class='hs-varid'>xs</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-513"></a><span class='hs-definition'>quasiShuffles</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-514"></a>
<a name="line-515"></a>
<a name="line-516"></a><a name="QSymM"></a><span class='hs-comment'>-- |A type for the monomial basis for the quasi-symmetric functions, indexed by compositions.</span>
<a name="line-517"></a><a name="QSymM"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-518"></a>
<a name="line-519"></a><a name="instance%20Ord%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-520"></a>    <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-521"></a>
<a name="line-522"></a><a name="instance%20Show%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-523"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"M "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>xs</span>
<a name="line-524"></a>
<a name="line-525"></a><a name="qsymM"></a><span class='hs-comment'>-- |Construct the element of QSym in the monomial basis indexed by the given composition</span>
<a name="line-526"></a><span class='hs-definition'>qsymM</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>QSymM</span>
<a name="line-527"></a><span class='hs-definition'>qsymM</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-528"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"qsymM: not a composition"</span>
<a name="line-529"></a>
<a name="line-530"></a><a name="instance%20Algebra%20k%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-531"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-532"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-533"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>,</span> <span class='hs-conid'>QSymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>gamma</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>gamma</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>quasiShuffles</span> <span class='hs-varid'>alpha</span> <span class='hs-varid'>beta</span><span class='hs-keyglyph'>]</span>
<a name="line-534"></a>
<a name="line-535"></a><a name="instance%20Coalgebra%20k%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-536"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>alpha</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-537"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-538"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>gamma</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>,</span> <span class='hs-conid'>QSymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>alpha</span><span class='hs-layout'>,</span><span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>deconcatenations</span> <span class='hs-varid'>gamma</span><span class='hs-keyglyph'>]</span>
<a name="line-539"></a>
<a name="line-540"></a><a name="instance%20Bialgebra%20k%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-541"></a>
<a name="line-542"></a><a name="instance%20HopfAlgebra%20k%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-543"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-544"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varop'>^</span><span class='hs-varid'>length</span> <span class='hs-varid'>alpha</span> <span class='hs-varop'>*</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>beta</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>coarsenings</span> <span class='hs-layout'>(</span><span class='hs-varid'>reverse</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-545"></a>        <span class='hs-comment'>-- antipode' (QSymM alpha) = (-1)^length alpha * sumv [return (QSymM (reverse beta)) | beta &lt;- coarsenings alpha]</span>
<a name="line-546"></a>
<a name="line-547"></a><a name="coarsenings"></a><span class='hs-definition'>coarsenings</span> <span class='hs-layout'>(</span><span class='hs-varid'>x1</span><span class='hs-conop'>:</span><span class='hs-varid'>x2</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>x1</span><span class='hs-conop'>:</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>coarsenings</span> <span class='hs-layout'>(</span><span class='hs-varid'>x2</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-varid'>coarsenings</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>x1</span><span class='hs-varop'>+</span><span class='hs-varid'>x2</span><span class='hs-layout'>)</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-548"></a><span class='hs-definition'>coarsenings</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- for xs a singleton or null</span>
<a name="line-549"></a>
<a name="line-550"></a><a name="refinements"></a><span class='hs-definition'>refinements</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>y</span><span class='hs-varop'>++</span><span class='hs-varid'>ys</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>compositions</span> <span class='hs-varid'>x</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>refinements</span> <span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-551"></a><span class='hs-definition'>refinements</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>[]</span><span class='hs-keyglyph'>]</span>
<a name="line-552"></a>
<a name="line-553"></a>
<a name="line-554"></a><a name="QSymF"></a><span class='hs-comment'>-- |A type for the fundamental basis for the quasi-symmetric functions, indexed by compositions.</span>
<a name="line-555"></a><a name="QSymF"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span>
<a name="line-556"></a>
<a name="line-557"></a><a name="instance%20Ord%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span>
<a name="line-558"></a>    <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span>
<a name="line-559"></a>
<a name="line-560"></a><a name="instance%20Show%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Show</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span>
<a name="line-561"></a>    <span class='hs-varid'>show</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-str'>"F "</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>xs</span>
<a name="line-562"></a>
<a name="line-563"></a><a name="qsymF"></a><span class='hs-comment'>-- |Construct the element of QSym in the fundamental basis indexed by the given composition</span>
<a name="line-564"></a><span class='hs-definition'>qsymF</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>QSymF</span>
<a name="line-565"></a><span class='hs-definition'>qsymF</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-566"></a>         <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"qsymF: not a composition"</span>
<a name="line-567"></a>
<a name="line-568"></a><a name="qsymMtoF"></a><span class='hs-comment'>-- |Convert an element of QSym represented in the monomial basis to the fundamental basis</span>
<a name="line-569"></a><span class='hs-definition'>qsymMtoF</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span>
<a name="line-570"></a><span class='hs-definition'>qsymMtoF</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>qsymMtoF'</span> <span class='hs-keyword'>where</span>
<a name="line-571"></a>    <span class='hs-varid'>qsymMtoF'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varop'>^</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>beta</span> <span class='hs-comment'>-</span> <span class='hs-varid'>length</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>beta</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>refinements</span> <span class='hs-varid'>alpha</span><span class='hs-keyglyph'>]</span>
<a name="line-572"></a>
<a name="line-573"></a><a name="qsymFtoM"></a><span class='hs-comment'>-- |Convert an element of QSym represented in the fundamental basis to the monomial basis</span>
<a name="line-574"></a><span class='hs-definition'>qsymFtoM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span>
<a name="line-575"></a><span class='hs-definition'>qsymFtoM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>qsymFtoM'</span> <span class='hs-keyword'>where</span>
<a name="line-576"></a>    <span class='hs-varid'>qsymFtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>beta</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>refinements</span> <span class='hs-varid'>alpha</span><span class='hs-keyglyph'>]</span> <span class='hs-comment'>-- ie beta &lt;- up-set of alpha</span>
<a name="line-577"></a>
<a name="line-578"></a><a name="instance%20Algebra%20k%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span>
<a name="line-579"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-580"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>qsymMtoF</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>qsymFtoM</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>qsymFtoM</span><span class='hs-layout'>)</span>
<a name="line-581"></a>
<a name="line-582"></a><a name="instance%20Coalgebra%20k%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span>
<a name="line-583"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-584"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>qsymMtoF</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>qsymMtoF</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>comult</span> <span class='hs-varop'>.</span> <span class='hs-varid'>qsymFtoM</span>
<a name="line-585"></a>
<a name="line-586"></a><a name="instance%20Bialgebra%20k%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-587"></a>
<a name="line-588"></a><a name="instance%20HopfAlgebra%20k%20QSymF"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span> <span class='hs-keyword'>where</span>
<a name="line-589"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>qsymMtoF</span> <span class='hs-varop'>.</span> <span class='hs-varid'>antipode</span> <span class='hs-varop'>.</span> <span class='hs-varid'>qsymFtoM</span>
<a name="line-590"></a>
<a name="line-591"></a>
<a name="line-592"></a><span class='hs-comment'>-- QUASI-SYMMETRIC POLYNOMIALS</span>
<a name="line-593"></a>
<a name="line-594"></a><span class='hs-comment'>-- the above induces Hopf algebra structure on quasi-symmetric functions via</span>
<a name="line-595"></a><span class='hs-comment'>-- m_alpha -&gt; sum [product (zipWith (^) (map x_ is) alpha | is &lt;- combinationsOf k [] ] where k = length alpha</span>
<a name="line-596"></a>
<a name="line-597"></a><span class='hs-comment'>-- xvars n = [glexvar ("x" ++ show i) | i &lt;- [1..n] ]</span>
<a name="line-598"></a>
<a name="line-599"></a><a name="qsymPoly"></a><span class='hs-comment'>-- |@qsymPoly n is@ is the quasi-symmetric polynomial in n variables for the indices is. (This corresponds to the</span>
<a name="line-600"></a><span class='hs-comment'>-- monomial basis for QSym.) For example, qsymPoly 3 [2,1] == x1^2*x2+x1^2*x3+x2^2*x3.</span>
<a name="line-601"></a><span class='hs-definition'>qsymPoly</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>GlexPoly</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>String</span>
<a name="line-602"></a><span class='hs-definition'>qsymPoly</span> <span class='hs-varid'>n</span> <span class='hs-varid'>is</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sum</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>product</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-varop'>^</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs'</span> <span class='hs-varid'>is</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>xs'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>combinationsOf</span> <span class='hs-varid'>r</span> <span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-603"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>glexvar</span> <span class='hs-layout'>(</span><span class='hs-str'>"x"</span> <span class='hs-varop'>++</span> <span class='hs-varid'>show</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-604"></a>          <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>is</span>
<a name="line-605"></a>
<a name="line-606"></a>
<a name="line-607"></a><span class='hs-comment'>-- SYM, THE HOPF ALGEBRA OF SYMMETRIC FUNCTIONS</span>
<a name="line-608"></a>
<a name="line-609"></a><a name="SymM"></a><span class='hs-comment'>-- |A type for the monomial basis for Sym, the Hopf algebra of symmetric functions, indexed by integer partitions</span>
<a name="line-610"></a><a name="SymM"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>SymM</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymM</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-611"></a>
<a name="line-612"></a><a name="instance%20Ord%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span>
<a name="line-613"></a>    <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>compare</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>sum</span> <span class='hs-varid'>ys</span><span class='hs-layout'>,</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-comment'>-- note the order reversal in snd</span>
<a name="line-614"></a>
<a name="line-615"></a><a name="symM"></a><span class='hs-comment'>-- |Construct the element of Sym in the monomial basis indexed by the given integer partition</span>
<a name="line-616"></a><span class='hs-definition'>symM</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>SymM</span>
<a name="line-617"></a><span class='hs-definition'>symM</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sortDesc</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-618"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"symM: not a partition"</span>
<a name="line-619"></a>
<a name="line-620"></a><a name="instance%20Algebra%20k%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span>
<a name="line-621"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-622"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-623"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>,</span> <span class='hs-conid'>SymM</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>nu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>nu</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>symMult</span> <span class='hs-varid'>lambda</span> <span class='hs-varid'>mu</span><span class='hs-keyglyph'>]</span>
<a name="line-624"></a>
<a name="line-625"></a><span class='hs-comment'>-- multisetPermutations = toSet . L.permutations</span>
<a name="line-626"></a>
<a name="line-627"></a><span class='hs-comment'>-- compositionsFromPartition2 = foldl (\xss ys -&gt; concatMap (shuffles ys) xss) [[]] . L.group</span>
<a name="line-628"></a><span class='hs-comment'>-- compositionsFromPartition2 = foldl (\ls r -&gt; concat [shuffles l r | l &lt;- ls]) [[]] . L.group</span>
<a name="line-629"></a>
<a name="line-630"></a><a name="compositionsFromPartition"></a><span class='hs-comment'>-- The partition must be in either ascending or descending order (so that L.group does as expected)</span>
<a name="line-631"></a><span class='hs-definition'>compositionsFromPartition</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldr</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>l</span> <span class='hs-varid'>rs</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>concatMap</span> <span class='hs-layout'>(</span><span class='hs-varid'>shuffles</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>[]</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>.</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>group</span>
<a name="line-632"></a>
<a name="line-633"></a><a name="symMult"></a><span class='hs-comment'>-- In effect, we multiply in Sym by converting to QSym, multiplying there, and converting back.</span>
<a name="line-634"></a><span class='hs-comment'>-- It would be nice to find a more direct method.</span>
<a name="line-635"></a><span class='hs-definition'>symMult</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>filter</span> <span class='hs-varid'>isWeaklyDecreasing</span> <span class='hs-varop'>$</span> <span class='hs-varid'>concat</span>
<a name="line-636"></a>    <span class='hs-keyglyph'>[</span><span class='hs-varid'>quasiShuffles</span> <span class='hs-varid'>xs'</span> <span class='hs-varid'>ys'</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>xs'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>compositionsFromPartition</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-varid'>ys'</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>compositionsFromPartition</span> <span class='hs-varid'>ys</span><span class='hs-keyglyph'>]</span>
<a name="line-637"></a>
<a name="line-638"></a><a name="instance%20Coalgebra%20k%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span>
<a name="line-639"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>lambda</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-640"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-641"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>mu</span><span class='hs-layout'>,</span> <span class='hs-conid'>SymM</span> <span class='hs-varid'>nu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>mu</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>toSet</span> <span class='hs-layout'>(</span><span class='hs-varid'>powersetdfs</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>nu</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>diffDesc</span> <span class='hs-varid'>lambda</span> <span class='hs-varid'>mu</span><span class='hs-keyglyph'>]</span>
<a name="line-642"></a>
<a name="line-643"></a><a name="instance%20Bialgebra%20k%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-644"></a>
<a name="line-645"></a><a name="instance%20HopfAlgebra%20k%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span>
<a name="line-646"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-647"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-648"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>negatev</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mult</span> <span class='hs-varop'>.</span> <span class='hs-layout'>(</span><span class='hs-varid'>id</span> <span class='hs-varop'>`tf`</span> <span class='hs-varid'>antipode</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>removeTerm</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-conid'>[]</span><span class='hs-layout'>,</span><span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>comult</span> <span class='hs-varop'>.</span> <span class='hs-varid'>return</span><span class='hs-layout'>)</span> <span class='hs-varid'>x</span>
<a name="line-649"></a>
<a name="line-650"></a>
<a name="line-651"></a><a name="SymE"></a><span class='hs-comment'>-- |The elementary basis for Sym, the Hopf algebra of symmetric functions. Defined informally as</span>
<a name="line-652"></a><a name="SymE"></a><span class='hs-comment'>-- &gt; symE [n] = symM (replicate n 1)</span>
<a name="line-653"></a><a name="SymE"></a><span class='hs-comment'>-- &gt; symE lambda = product [symE [p] | p &lt;- lambda]</span>
<a name="line-654"></a><a name="SymE"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span><span class='hs-conid'>Ord</span><span class='hs-layout'>,</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-655"></a>
<a name="line-656"></a><a name="symE"></a><span class='hs-definition'>symE</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>SymE</span>
<a name="line-657"></a><span class='hs-definition'>symE</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sortDesc</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-658"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"symE: not a partition"</span>
<a name="line-659"></a>
<a name="line-660"></a><a name="instance%20Algebra%20k%20SymE"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymE</span> <span class='hs-keyword'>where</span>
<a name="line-661"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-662"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>,</span> <span class='hs-conid'>SymE</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>SymE</span> <span class='hs-varop'>$</span> <span class='hs-varid'>multisetSumDesc</span> <span class='hs-varid'>lambda</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span>
<a name="line-663"></a>
<a name="line-664"></a><a name="instance%20Coalgebra%20k%20SymE"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymE</span> <span class='hs-keyword'>where</span>
<a name="line-665"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>lambda</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-666"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-667"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>e</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>e</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-668"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>lambda</span><span class='hs-keyglyph'>]</span>
<a name="line-669"></a>        <span class='hs-varid'>e</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymE</span> <span class='hs-conid'>[]</span>
<a name="line-670"></a>        <span class='hs-varid'>e</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>i</span><span class='hs-keyglyph'>]</span>
<a name="line-671"></a>
<a name="line-672"></a><a name="instance%20Bialgebra%20k%20SymE"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymE</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-673"></a>
<a name="line-674"></a><a name="symEtoM"></a><span class='hs-comment'>-- |Convert from the elementary to the monomial basis of Sym</span>
<a name="line-675"></a><span class='hs-definition'>symEtoM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span>
<a name="line-676"></a><span class='hs-definition'>symEtoM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>symEtoM'</span> <span class='hs-keyword'>where</span>
<a name="line-677"></a>    <span class='hs-varid'>symEtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-layout'>(</span><span class='hs-varid'>replicate</span> <span class='hs-varid'>n</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-678"></a>    <span class='hs-varid'>symEtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>symEtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymE</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>p</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>lambda</span><span class='hs-keyglyph'>]</span>
<a name="line-679"></a>
<a name="line-680"></a>
<a name="line-681"></a><a name="SymH"></a><span class='hs-comment'>-- |The complete basis for Sym, the Hopf algebra of symmetric functions. Defined informally as</span>
<a name="line-682"></a><a name="SymH"></a><span class='hs-comment'>-- &gt; symH [n] = sum [symM lambda | lambda &lt;- integerPartitions n] -- == all monomials of weight n</span>
<a name="line-683"></a><a name="SymH"></a><span class='hs-comment'>-- &gt; symH lambda = product [symH [p] | p &lt;- lambda]</span>
<a name="line-684"></a><a name="SymH"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span><span class='hs-conid'>Ord</span><span class='hs-layout'>,</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-685"></a>
<a name="line-686"></a><a name="symH"></a><span class='hs-definition'>symH</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>SymH</span>
<a name="line-687"></a><span class='hs-definition'>symH</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sortDesc</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-688"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"symH: not a partition"</span>
<a name="line-689"></a>
<a name="line-690"></a><a name="instance%20Algebra%20k%20SymH"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span> <span class='hs-keyword'>where</span>
<a name="line-691"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-692"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>,</span> <span class='hs-conid'>SymH</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>SymH</span> <span class='hs-varop'>$</span> <span class='hs-varid'>multisetSumDesc</span> <span class='hs-varid'>lambda</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span>
<a name="line-693"></a>
<a name="line-694"></a><a name="instance%20Coalgebra%20k%20SymH"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span> <span class='hs-keyword'>where</span>
<a name="line-695"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>lambda</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-696"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-697"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>h</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-698"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>lambda</span><span class='hs-keyglyph'>]</span>
<a name="line-699"></a>        <span class='hs-varid'>h</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymH</span> <span class='hs-conid'>[]</span>
<a name="line-700"></a>        <span class='hs-varid'>h</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>i</span><span class='hs-keyglyph'>]</span>
<a name="line-701"></a>
<a name="line-702"></a><a name="instance%20Bialgebra%20k%20SymH"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-703"></a>
<a name="line-704"></a><a name="symHtoM"></a><span class='hs-comment'>-- |Convert from the complete to the monomial basis of Sym</span>
<a name="line-705"></a><span class='hs-definition'>symHtoM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span>
<a name="line-706"></a><span class='hs-definition'>symHtoM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>symHtoM'</span> <span class='hs-keyword'>where</span>
<a name="line-707"></a>    <span class='hs-varid'>symHtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>mu</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>mu</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>integerPartitions</span> <span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span>
<a name="line-708"></a>    <span class='hs-varid'>symHtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varid'>lambda</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>symHtoM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>p</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>lambda</span><span class='hs-keyglyph'>]</span>
<a name="line-709"></a>
<a name="line-710"></a>
<a name="line-711"></a><span class='hs-comment'>-- NSYM, THE HOPF ALGEBRA OF NON-COMMUTATIVE SYMMETRIC FUNCTIONS</span>
<a name="line-712"></a>
<a name="line-713"></a><a name="NSym"></a><span class='hs-comment'>-- |A basis for NSym, the Hopf algebra of non-commutative symmetric functions, indexed by compositions</span>
<a name="line-714"></a><a name="NSym"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>,</span><span class='hs-conid'>Ord</span><span class='hs-layout'>,</span><span class='hs-conid'>Show</span><span class='hs-layout'>)</span>
<a name="line-715"></a>
<a name="line-716"></a><a name="nsym"></a><span class='hs-definition'>nsym</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>Int</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-conid'>Q</span> <span class='hs-conid'>NSym</span>
<a name="line-717"></a><span class='hs-definition'>nsym</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-718"></a><span class='hs-definition'>nsym</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>all</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-719"></a>        <span class='hs-keyglyph'>|</span> <span class='hs-varid'>otherwise</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"nsym: not a composition"</span>
<a name="line-720"></a>
<a name="line-721"></a><a name="instance%20Algebra%20k%20NSym"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Algebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-keyword'>where</span>
<a name="line-722"></a>    <span class='hs-varid'>unit</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span>
<a name="line-723"></a>    <span class='hs-varid'>mult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>mult'</span> <span class='hs-keyword'>where</span>
<a name="line-724"></a>        <span class='hs-varid'>mult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span> <span class='hs-conid'>NSym</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varop'>$</span> <span class='hs-conid'>NSym</span> <span class='hs-varop'>$</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>ys</span>
<a name="line-725"></a>
<a name="line-726"></a><a name="instance%20Coalgebra%20k%20NSym"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Coalgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-keyword'>where</span>
<a name="line-727"></a>    <span class='hs-varid'>counit</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>unwrap</span> <span class='hs-varop'>.</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>counit'</span> <span class='hs-keyword'>where</span> <span class='hs-varid'>counit'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>null</span> <span class='hs-varid'>zs</span> <span class='hs-keyword'>then</span> <span class='hs-num'>1</span> <span class='hs-keyword'>else</span> <span class='hs-num'>0</span>
<a name="line-728"></a>    <span class='hs-varid'>comult</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>comult'</span> <span class='hs-keyword'>where</span>
<a name="line-729"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>z</span> <span class='hs-varid'>i</span><span class='hs-layout'>,</span> <span class='hs-varid'>z</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>0</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>]</span>
<a name="line-730"></a>        <span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>comult'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>zs</span><span class='hs-keyglyph'>]</span>
<a name="line-731"></a>        <span class='hs-varid'>z</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NSym</span> <span class='hs-conid'>[]</span>
<a name="line-732"></a>        <span class='hs-varid'>z</span> <span class='hs-varid'>i</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>i</span><span class='hs-keyglyph'>]</span>
<a name="line-733"></a>
<a name="line-734"></a><a name="instance%20Bialgebra%20k%20NSym"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Bialgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-keyword'>where</span> <span class='hs-layout'>{</span><span class='hs-layout'>}</span>
<a name="line-735"></a>
<a name="line-736"></a><a name="instance%20HopfAlgebra%20k%20NSym"></a><span class='hs-comment'>-- Hazewinkel et al p233</span>
<a name="line-737"></a><a name="instance%20HopfAlgebra%20k%20NSym"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HopfAlgebra</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-keyword'>where</span>
<a name="line-738"></a>    <span class='hs-varid'>antipode</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>antipode'</span> <span class='hs-keyword'>where</span>
<a name="line-739"></a>        <span class='hs-varid'>antipode'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span><span class='hs-varop'>^</span><span class='hs-varid'>length</span> <span class='hs-varid'>beta</span> <span class='hs-varop'>*&gt;</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>beta</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>refinements</span> <span class='hs-layout'>(</span><span class='hs-varid'>reverse</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>)</span><span class='hs-keyglyph'>]</span>
<a name="line-740"></a>
<a name="line-741"></a>
<a name="line-742"></a>
<a name="line-743"></a><span class='hs-comment'>-- MAPS BETWEEN (POSETS AND) HOPF ALGEBRAS</span>
<a name="line-744"></a>
<a name="line-745"></a><a name="descendingTree"></a><span class='hs-comment'>-- A descending tree is one in which a child is always less than a parent.</span>
<a name="line-746"></a><span class='hs-definition'>descendingTree</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span>
<a name="line-747"></a><span class='hs-definition'>descendingTree</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-conid'>E</span> <span class='hs-varid'>x</span> <span class='hs-conid'>E</span>
<a name="line-748"></a><span class='hs-definition'>descendingTree</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span>
<a name="line-749"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>x</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>maximum</span> <span class='hs-varid'>xs</span>
<a name="line-750"></a>          <span class='hs-layout'>(</span><span class='hs-varid'>ls</span><span class='hs-layout'>,</span><span class='hs-keyword'>_</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>break</span> <span class='hs-layout'>(</span><span class='hs-varop'>==</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span>
<a name="line-751"></a>          <span class='hs-varid'>l</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>descendingTree</span> <span class='hs-varid'>ls</span>
<a name="line-752"></a>          <span class='hs-varid'>r</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>descendingTree</span> <span class='hs-varid'>rs</span>
<a name="line-753"></a><span class='hs-comment'>-- This is a bijection from permutations to "ordered trees".</span>
<a name="line-754"></a><span class='hs-comment'>-- It is order-preserving on trees with the same nodecount.</span>
<a name="line-755"></a><span class='hs-comment'>-- We can recover the permutation by reading the node labels in infix order.</span>
<a name="line-756"></a><span class='hs-comment'>-- This is the map called lambda in Loday.pdf</span>
<a name="line-757"></a>
<a name="line-758"></a>
<a name="line-759"></a><a name="descendingTreeMap"></a><span class='hs-comment'>-- |Given a permutation p of [1..n], we can construct a tree (the descending tree of p) as follows:</span>
<a name="line-760"></a><span class='hs-comment'>--</span>
<a name="line-761"></a><span class='hs-comment'>-- * Split the permutation as p = ls ++ [n] ++ rs</span>
<a name="line-762"></a><span class='hs-comment'>--</span>
<a name="line-763"></a><span class='hs-comment'>-- * Place n at the root of the tree, and recursively place the descending trees of ls and rs as the left and right children of the root</span>
<a name="line-764"></a><span class='hs-comment'>--</span>
<a name="line-765"></a><span class='hs-comment'>-- * To bottom out the recursion, the descending tree of the empty permutation is of course the empty tree</span>
<a name="line-766"></a><span class='hs-comment'>--</span>
<a name="line-767"></a><span class='hs-comment'>-- This map between bases SSymF -&gt; YSymF turns out to induce a morphism of Hopf algebras.</span>
<a name="line-768"></a><span class='hs-definition'>descendingTreeMap</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-conid'>()</span><span class='hs-layout'>)</span>
<a name="line-769"></a><span class='hs-definition'>descendingTreeMap</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varop'>.</span> <span class='hs-varid'>shape</span> <span class='hs-varop'>.</span>  <span class='hs-varid'>descendingTree'</span><span class='hs-layout'>)</span>
<a name="line-770"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>descendingTree'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>descendingTree</span> <span class='hs-varid'>xs</span>
<a name="line-771"></a><span class='hs-comment'>-- This is the map called Lambda in Loday.pdf, or tau in MSym.pdf</span>
<a name="line-772"></a><span class='hs-comment'>-- It is an algebra morphism.</span>
<a name="line-773"></a>
<a name="line-774"></a><span class='hs-comment'>-- One of the ideas in the MSym paper is to look at the intermediate result (fmap descendingTree' x),</span>
<a name="line-775"></a><span class='hs-comment'>-- which is an "ordered tree", and consider the map as factored through this</span>
<a name="line-776"></a>
<a name="line-777"></a><span class='hs-comment'>-- The map is surjective but not injective. The fibers tau^-1(t) are intervals in the weak order on permutations</span>
<a name="line-778"></a>
<a name="line-779"></a><a name="minPerm"></a><span class='hs-comment'>-- "inverse" for descendingTree</span>
<a name="line-780"></a><span class='hs-comment'>-- These are the maps called gamma in Loday.pdf</span>
<a name="line-781"></a><span class='hs-comment'>-- or are they? - these give the min and max inverse images in the lexicographic order, rather than the weak order?</span>
<a name="line-782"></a><span class='hs-definition'>minPerm</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>minPerm'</span> <span class='hs-layout'>(</span><span class='hs-varid'>lrCountTree</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-783"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>minPerm'</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-784"></a>          <span class='hs-varid'>minPerm'</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-layout'>(</span><span class='hs-varid'>lc</span><span class='hs-layout'>,</span><span class='hs-varid'>rc</span><span class='hs-layout'>)</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>minPerm'</span> <span class='hs-varid'>l</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>lc</span><span class='hs-varop'>+</span><span class='hs-varid'>rc</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-varid'>lc</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>minPerm'</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span>
<a name="line-785"></a>
<a name="line-786"></a><a name="maxPerm"></a><span class='hs-definition'>maxPerm</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>maxPerm'</span> <span class='hs-layout'>(</span><span class='hs-varid'>lrCountTree</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-787"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>maxPerm'</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-788"></a>          <span class='hs-varid'>maxPerm'</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-layout'>(</span><span class='hs-varid'>lc</span><span class='hs-layout'>,</span><span class='hs-varid'>rc</span><span class='hs-layout'>)</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-varid'>rc</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>maxPerm'</span> <span class='hs-varid'>l</span><span class='hs-layout'>)</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>lc</span><span class='hs-varop'>+</span><span class='hs-varid'>rc</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span> <span class='hs-varop'>++</span> <span class='hs-varid'>maxPerm'</span> <span class='hs-varid'>r</span>
<a name="line-789"></a>
<a name="line-790"></a>
<a name="line-791"></a><a name="leftLeafComposition"></a><span class='hs-comment'>-- The composition of [1..n] obtained by treating each left-facing leaf as a cut</span>
<a name="line-792"></a><span class='hs-comment'>-- Specifically, we visit the nodes in infix order, cutting after a node if it does not have an E as its right child</span>
<a name="line-793"></a><span class='hs-comment'>-- This is the map called L in Loday.pdf</span>
<a name="line-794"></a><span class='hs-definition'>leftLeafComposition</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-795"></a><span class='hs-definition'>leftLeafComposition</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>cuts</span> <span class='hs-varop'>$</span> <span class='hs-varid'>tail</span> <span class='hs-varop'>$</span> <span class='hs-varid'>leftLeafs</span> <span class='hs-varid'>t</span>
<a name="line-796"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>leftLeafs</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leftLeafs</span> <span class='hs-varid'>l</span> <span class='hs-varop'>++</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>False</span><span class='hs-keyglyph'>]</span>
<a name="line-797"></a>          <span class='hs-varid'>leftLeafs</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leftLeafs</span> <span class='hs-varid'>l</span> <span class='hs-varop'>++</span> <span class='hs-varid'>leftLeafs</span> <span class='hs-varid'>r</span>
<a name="line-798"></a>          <span class='hs-varid'>leftLeafs</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>True</span><span class='hs-keyglyph'>]</span>
<a name="line-799"></a>          <span class='hs-varid'>cuts</span> <span class='hs-varid'>bs</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>case</span> <span class='hs-varid'>break</span> <span class='hs-varid'>id</span> <span class='hs-varid'>bs</span> <span class='hs-keyword'>of</span>
<a name="line-800"></a>                    <span class='hs-layout'>(</span><span class='hs-varid'>ls</span><span class='hs-layout'>,</span><span class='hs-varid'>r</span><span class='hs-conop'>:</span><span class='hs-varid'>rs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-varid'>length</span> <span class='hs-varid'>ls</span> <span class='hs-varop'>+</span> <span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>cuts</span> <span class='hs-varid'>rs</span>
<a name="line-801"></a>                    <span class='hs-layout'>(</span><span class='hs-varid'>ls</span><span class='hs-layout'>,</span><span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>length</span> <span class='hs-varid'>ls</span><span class='hs-keyglyph'>]</span>
<a name="line-802"></a>
<a name="line-803"></a><a name="leftLeafComposition'"></a><span class='hs-definition'>leftLeafComposition'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>QSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>leftLeafComposition</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-804"></a>
<a name="line-805"></a><a name="leftLeafCompositionMap"></a><span class='hs-comment'>-- |A Hopf algebra morphism from YSymF to QSymF</span>
<a name="line-806"></a><span class='hs-definition'>leftLeafCompositionMap</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymF</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span>
<a name="line-807"></a><span class='hs-definition'>leftLeafCompositionMap</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>leftLeafComposition'</span>
<a name="line-808"></a>
<a name="line-809"></a>
<a name="line-810"></a><a name="descents"></a><span class='hs-comment'>-- The descent set of a permutation is [i | x_i &gt; x_i+1], where we start the indexing from 1</span>
<a name="line-811"></a><span class='hs-definition'>descents</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-812"></a><span class='hs-definition'>descents</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-varop'>$</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>elemIndices</span> <span class='hs-conid'>True</span> <span class='hs-varop'>$</span> <span class='hs-varid'>zipWith</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-layout'>(</span><span class='hs-varid'>tail</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-813"></a>
<a name="line-814"></a><a name="descentComposition"></a><span class='hs-comment'>-- The composition of [1..n] obtained by treating each descent as a cut</span>
<a name="line-815"></a><span class='hs-definition'>descentComposition</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-816"></a><span class='hs-definition'>descentComposition</span> <span class='hs-varid'>xs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>descComp</span> <span class='hs-num'>0</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>where</span>
<a name="line-817"></a>    <span class='hs-varid'>descComp</span> <span class='hs-varid'>c</span> <span class='hs-layout'>(</span><span class='hs-varid'>x1</span><span class='hs-conop'>:</span><span class='hs-varid'>x2</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x1</span> <span class='hs-varop'>&lt;</span> <span class='hs-varid'>x2</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>descComp</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>x2</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyword'>else</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <span class='hs-conop'>:</span> <span class='hs-varid'>descComp</span> <span class='hs-num'>0</span> <span class='hs-layout'>(</span><span class='hs-varid'>x2</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span>
<a name="line-818"></a>    <span class='hs-varid'>descComp</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>x</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>c</span><span class='hs-varop'>+</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span>
<a name="line-819"></a>
<a name="line-820"></a><a name="descentMap"></a><span class='hs-comment'>-- |Given a permutation of [1..n], its descents are those positions where the next number is less than the previous number.</span>
<a name="line-821"></a><span class='hs-comment'>-- For example, the permutation [2,3,5,1,6,4] has descents from 5 to 1 and from 6 to 4. The descents can be regarded as cutting</span>
<a name="line-822"></a><span class='hs-comment'>-- the permutation sequence into segments - 235-16-4 - and by counting the lengths of the segments, we get a composition 3+2+1.</span>
<a name="line-823"></a><span class='hs-comment'>-- This map between bases SSymF -&gt; QSymF turns out to induce a morphism of Hopf algebras.</span>
<a name="line-824"></a><span class='hs-definition'>descentMap</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymF</span>
<a name="line-825"></a><span class='hs-definition'>descentMap</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>nf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>QSymF</span> <span class='hs-layout'>(</span><span class='hs-varid'>descentComposition</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-826"></a><span class='hs-comment'>-- descentMap == leftLeafCompositionMap . descendingTreeMap</span>
<a name="line-827"></a>
<a name="line-828"></a><a name="underComposition"></a><span class='hs-definition'>underComposition</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymF</span> <span class='hs-varid'>ps</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>foldr</span> <span class='hs-varid'>under</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>[</span><span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>p</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>ps</span><span class='hs-keyglyph'>]</span>
<a name="line-829"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>under</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>let</span> <span class='hs-varid'>q</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>length</span> <span class='hs-varid'>ys</span>
<a name="line-830"></a>                                            <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-varid'>q</span><span class='hs-layout'>)</span> <span class='hs-varid'>xs</span> <span class='hs-varop'>++</span> <span class='hs-varid'>ys</span> <span class='hs-comment'>-- so it has a global descent at the split</span>
<a name="line-831"></a>                                        <span class='hs-keyword'>in</span> <span class='hs-conid'>SSymF</span> <span class='hs-varid'>zs</span>
<a name="line-832"></a><span class='hs-comment'>-- This is a poset morphism (indeed, it forms a Galois connection with descentComposition)</span>
<a name="line-833"></a><span class='hs-comment'>-- but it does not extend to a Hopf algebra morphism.</span>
<a name="line-834"></a><span class='hs-comment'>-- (It does extend to a coalgebra morphism.)</span>
<a name="line-835"></a><span class='hs-comment'>-- (It is picking the maximum permutation having a given descent composition,</span>
<a name="line-836"></a><span class='hs-comment'>-- so there's an element of arbitrariness to it.)</span>
<a name="line-837"></a><span class='hs-comment'>-- This is the map called Z (Zeta?) in Loday.pdf</span>
<a name="line-838"></a>
<a name="line-839"></a><span class='hs-comment'>{-
<a name="line-840"></a>-- This is O(n^2), whereas an O(n) implementation should be possible
<a name="line-841"></a>-- Also, we would really like the associated composition (obtained by treating each global descent as a cut)?
<a name="line-842"></a>globalDescents xs = globalDescents' 0 [] xs
<a name="line-843"></a>    where globalDescents' i ls (r:rs) = (if minimum (infinity:ls) &gt; maximum (0:r:rs) then [i] else [])
<a name="line-844"></a>                                     ++ globalDescents' (i+1) (r:ls) rs
<a name="line-845"></a>          globalDescents' n _ [] = [n]
<a name="line-846"></a>          infinity = maxBound :: Int
<a name="line-847"></a>-- The idea is that this leads to a map from SSymM to QSymM
<a name="line-848"></a>
<a name="line-849"></a>globalDescentComposition [] = []
<a name="line-850"></a>globalDescentComposition (x:xs) = globalDescents' 1 x xs
<a name="line-851"></a>    where globalDescents' i minl (r:rs) = if minl &gt; maximum (r:rs)
<a name="line-852"></a>                                          then i : globalDescents' 1 r rs
<a name="line-853"></a>                                          else globalDescents' (i+1) r rs
<a name="line-854"></a>          globalDescents' i _ [] = [i]
<a name="line-855"></a>
<a name="line-856"></a>globalDescentMap :: (Eq k, Num k) =&gt; Vect k SSymM -&gt; Vect k QSymM
<a name="line-857"></a>globalDescentMap = nf . fmap (\(SSymM xs) -&gt; QSymM (globalDescentComposition xs))
<a name="line-858"></a>-}</span>
<a name="line-859"></a>
<a name="line-860"></a><a name="under"></a><span class='hs-comment'>-- A multiplication operation on trees</span>
<a name="line-861"></a><span class='hs-comment'>-- (Connected with their being cofree)</span>
<a name="line-862"></a><span class='hs-comment'>-- (intended to be used as infix)</span>
<a name="line-863"></a><span class='hs-definition'>under</span> <span class='hs-conid'>E</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>t</span>
<a name="line-864"></a><span class='hs-definition'>under</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>t</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-layout'>(</span><span class='hs-varid'>under</span> <span class='hs-varid'>r</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-865"></a>
<a name="line-866"></a><a name="isUnderIrreducible"></a><span class='hs-definition'>isUnderIrreducible</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-conid'>E</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>True</span>
<a name="line-867"></a><span class='hs-definition'>isUnderIrreducible</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>False</span>
<a name="line-868"></a>
<a name="line-869"></a><a name="underDecomposition"></a><span class='hs-definition'>underDecomposition</span> <span class='hs-layout'>(</span><span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>T</span> <span class='hs-varid'>l</span> <span class='hs-varid'>x</span> <span class='hs-conid'>E</span> <span class='hs-conop'>:</span> <span class='hs-varid'>underDecomposition</span> <span class='hs-varid'>r</span>
<a name="line-870"></a><span class='hs-definition'>underDecomposition</span> <span class='hs-conid'>E</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span>
<a name="line-871"></a>
<a name="line-872"></a>
<a name="line-873"></a><a name="ysymmToSh"></a><span class='hs-comment'>-- GHC7.4.1 doesn't like the following type signature - a bug.</span>
<a name="line-874"></a><span class='hs-comment'>-- ysymmToSh :: (Eq k, Num k) =&gt; Vect k (YSymM) =&gt; Vect k (Shuffle (PBT ()))</span>
<a name="line-875"></a><span class='hs-definition'>ysymmToSh</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>ysymmToSh'</span>
<a name="line-876"></a>    <span class='hs-keyword'>where</span> <span class='hs-varid'>ysymmToSh'</span> <span class='hs-layout'>(</span><span class='hs-conid'>YSymM</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Sh</span> <span class='hs-layout'>(</span><span class='hs-varid'>underDecomposition</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-877"></a><span class='hs-comment'>-- This is a coalgebra morphism (but not an algebra morphism)</span>
<a name="line-878"></a><span class='hs-comment'>-- It shows that YSym is co-free</span>
<a name="line-879"></a><span class='hs-comment'>{-
<a name="line-880"></a>-- This one not working yet - perhaps it needs an nf, or to go via S/YSymF, or ...
<a name="line-881"></a>ssymmToSh = nf . fmap ssymmToSh'
<a name="line-882"></a>    where ssymmToSh' (SSymM xs) = (Sh . underDecomposition . shape . descendingTree) xs
<a name="line-883"></a>-}</span>
<a name="line-884"></a>
<a name="line-885"></a><a name="symToQSymM"></a><span class='hs-comment'>-- |The injection of Sym into QSym (defined over the monomial basis)</span>
<a name="line-886"></a><span class='hs-definition'>symToQSymM</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymM</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>QSymM</span>
<a name="line-887"></a><span class='hs-definition'>symToQSymM</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>symToQSymM'</span> <span class='hs-keyword'>where</span>
<a name="line-888"></a>    <span class='hs-varid'>symToQSymM'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymM</span> <span class='hs-varid'>ps</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>sumv</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>QSymM</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>compositionsFromPartition</span> <span class='hs-varid'>ps</span><span class='hs-keyglyph'>]</span>
<a name="line-889"></a>
<a name="line-890"></a><a name="nsymToSymH"></a><span class='hs-comment'>-- We could equally well send NSym -&gt; SymE, since the algebra and coalgebra definitions for SymE and SymH are exactly analogous.</span>
<a name="line-891"></a><span class='hs-comment'>-- However, NSym -&gt; SymH is more natural, since it is consistent with the duality pairings below.</span>
<a name="line-892"></a><span class='hs-comment'>-- eg Hazewinkel 238ff</span>
<a name="line-893"></a><span class='hs-comment'>-- (Why do SymE and SymH have the same definitions? They're not dual bases. It's because of the Wronski relations.)</span>
<a name="line-894"></a><span class='hs-comment'>-- |A surjection of NSym onto Sym (defined over the complete basis)</span>
<a name="line-895"></a><span class='hs-definition'>nsymToSymH</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Vect</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span>
<a name="line-896"></a><span class='hs-definition'>nsymToSymH</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>nsymToSym'</span> <span class='hs-keyword'>where</span>
<a name="line-897"></a>    <span class='hs-varid'>nsymToSym'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varop'>$</span> <span class='hs-varid'>sortDesc</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span>
<a name="line-898"></a>
<a name="line-899"></a><a name="nsymToSSym"></a><span class='hs-comment'>-- The Hopf algebra morphism NSym -&gt; Sym factors through NSym -&gt; SSym -&gt; YSym -&gt; Sym (contained in QSym)</span>
<a name="line-900"></a><span class='hs-comment'>-- (?? This map NSym -&gt; SSym is the dual of the descent map SSym -&gt; QSym ??)</span>
<a name="line-901"></a><span class='hs-comment'>-- (Loday.pdf, p30)</span>
<a name="line-902"></a><span class='hs-comment'>-- (See also Hazewinkel p267-9)</span>
<a name="line-903"></a><span class='hs-definition'>nsymToSSym</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>nsymToSSym'</span> <span class='hs-keyword'>where</span>
<a name="line-904"></a>    <span class='hs-varid'>nsymToSSym'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>product</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-conid'>SSymF</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>..</span><span class='hs-varid'>n</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>|</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-varid'>xs</span><span class='hs-keyglyph'>]</span>
<a name="line-905"></a>
<a name="line-906"></a>
<a name="line-907"></a><a name="instance%20HasPairing%20k%20SymH%20SymM"></a><span class='hs-comment'>-- |A duality pairing between the complete and monomial bases of Sym, showing that Sym is self-dual.</span>
<a name="line-908"></a><a name="instance%20HasPairing%20k%20SymH%20SymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HasPairing</span> <span class='hs-varid'>k</span> <span class='hs-conid'>SymH</span> <span class='hs-conid'>SymM</span> <span class='hs-keyword'>where</span>
<a name="line-909"></a>    <span class='hs-varid'>pairing</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>pairing'</span> <span class='hs-keyword'>where</span>
<a name="line-910"></a>        <span class='hs-varid'>pairing'</span> <span class='hs-layout'>(</span><span class='hs-conid'>SymH</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>,</span> <span class='hs-conid'>SymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>delta</span> <span class='hs-varid'>alpha</span> <span class='hs-varid'>beta</span> <span class='hs-comment'>-- Kronecker delta</span>
<a name="line-911"></a><span class='hs-comment'>-- Hazewinkel p178</span>
<a name="line-912"></a><span class='hs-comment'>-- Actually to show duality you would need to show that the map SymH -&gt; SymM*, v -&gt; &lt;v,.&gt; is onto</span>
<a name="line-913"></a>
<a name="line-914"></a><a name="instance%20HasPairing%20k%20NSym%20QSymM"></a><span class='hs-comment'>-- |A duality pairing between NSym and QSymM (monomial basis), showing that NSym and QSym are dual.</span>
<a name="line-915"></a><a name="instance%20HasPairing%20k%20NSym%20QSymM"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>HasPairing</span> <span class='hs-varid'>k</span> <span class='hs-conid'>NSym</span> <span class='hs-conid'>QSymM</span> <span class='hs-keyword'>where</span>
<a name="line-916"></a>    <span class='hs-varid'>pairing</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>linear</span> <span class='hs-varid'>pairing'</span> <span class='hs-keyword'>where</span>
<a name="line-917"></a>        <span class='hs-varid'>pairing'</span> <span class='hs-layout'>(</span><span class='hs-conid'>NSym</span> <span class='hs-varid'>alpha</span><span class='hs-layout'>,</span> <span class='hs-conid'>QSymM</span> <span class='hs-varid'>beta</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>delta</span> <span class='hs-varid'>alpha</span> <span class='hs-varid'>beta</span> <span class='hs-comment'>-- Kronecker delta</span>
<a name="line-918"></a><span class='hs-comment'>-- Hazewinkel p236-7</span>
<a name="line-919"></a><span class='hs-comment'>-- Actually to show duality you would need to show that the map NSym -&gt; QSymM*, v -&gt; &lt;v,.&gt; is onto</span>
<a name="line-920"></a>
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