HaskellForMaths-0.4.9: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Combinatorics.CombinatorialHopfAlgebra

Description

A module defining the following Combinatorial Hopf Algebras, together with coalgebra or Hopf algebra morphisms between them:

• Sh, the Shuffle Hopf algebra
• SSym, the Malvenuto-Reutnenauer Hopf algebra of permutations
• YSym, the (dual of the) Loday-Ronco Hopf algebra of binary trees
• QSym, the Hopf algebra of quasi-symmetric functions (having a basis indexed by compositions)
• Sym, the Hopf algebra of symmetric functions (having a basis indexed by integer partitions)
• NSym, the Hopf algebra of non-commutative symmetric functions
Synopsis

Documentation

class Graded b where Source #

Methods

grade :: b -> Int Source #

Instances
 Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Source # Instance details Methods Graded b => Graded (Dual b) Source # Instance details Methodsgrade :: Dual b -> Int Source # Graded (YSymF a) Source # Instance details Methodsgrade :: YSymF a -> Int Source # Source # Instance details Methodsgrade :: Shuffle a -> Int Source #

class (Eq k, Num k, Ord b, Graded b, HopfAlgebra k b) => CombinatorialHopfAlgebra k b where Source #

Methods

zeta :: Vect k b -> Vect k () Source #

gradedConnectedAntipode :: (Eq k, Num k, Ord b, Bialgebra k b, Graded b) => Vect k b -> Vect k b Source #

newtype Shuffle a Source #

A basis for the shuffle algebra. As a vector space, the shuffle algebra is identical to the tensor algebra. However, we consider a different algebra structure, based on the shuffle product. Together with the deconcatenation coproduct, this leads to a Hopf algebra structure.

Constructors

 Sh [a]
Instances
 (Eq k, Num k, Ord a) => HopfAlgebra k (Shuffle a) Source # Instance details Methodsantipode :: Vect k (Shuffle a) -> Vect k (Shuffle a) Source # (Eq k, Num k, Ord a) => Bialgebra k (Shuffle a) Source # Instance details (Eq k, Num k, Ord a) => Coalgebra k (Shuffle a) Source # Instance details Methodscounit :: Vect k (Shuffle a) -> k Source #comult :: Vect k (Shuffle a) -> Vect k (Tensor (Shuffle a) (Shuffle a)) Source # (Eq k, Num k, Ord a) => Algebra k (Shuffle a) Source # Instance details Methodsunit :: k -> Vect k (Shuffle a) Source #mult :: Vect k (Tensor (Shuffle a) (Shuffle a)) -> Vect k (Shuffle a) Source # Eq a => Eq (Shuffle a) Source # Instance details Methods(==) :: Shuffle a -> Shuffle a -> Bool #(/=) :: Shuffle a -> Shuffle a -> Bool # Ord a => Ord (Shuffle a) Source # Instance details Methodscompare :: Shuffle a -> Shuffle a -> Ordering #(<) :: Shuffle a -> Shuffle a -> Bool #(<=) :: Shuffle a -> Shuffle a -> Bool #(>) :: Shuffle a -> Shuffle a -> Bool #(>=) :: Shuffle a -> Shuffle a -> Bool #max :: Shuffle a -> Shuffle a -> Shuffle a #min :: Shuffle a -> Shuffle a -> Shuffle a # Show a => Show (Shuffle a) Source # Instance details MethodsshowsPrec :: Int -> Shuffle a -> ShowS #show :: Shuffle a -> String #showList :: [Shuffle a] -> ShowS # Source # Instance details Methodsgrade :: Shuffle a -> Int Source #

sh :: [a] -> Vect Q (Shuffle a) Source #

Construct a basis element of the shuffle algebra

shuffles :: [a] -> [a] -> [[a]] Source #

deconcatenations :: [a] -> [([a], [a])] Source #

newtype SSymF Source #

The fundamental basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym.

Constructors

 SSymF [Int]
Instances
 Source # Instance details Methods(==) :: SSymF -> SSymF -> Bool #(/=) :: SSymF -> SSymF -> Bool # Source # Instance details Methods(<) :: SSymF -> SSymF -> Bool #(<=) :: SSymF -> SSymF -> Bool #(>) :: SSymF -> SSymF -> Bool #(>=) :: SSymF -> SSymF -> Bool #max :: SSymF -> SSymF -> SSymF #min :: SSymF -> SSymF -> SSymF # Source # Instance details MethodsshowsPrec :: Int -> SSymF -> ShowS #show :: SSymF -> String #showList :: [SSymF] -> ShowS # Source # Instance details Methods Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k SSymF Source # Instance details Methods (Eq k, Num k) => Bialgebra k SSymF Source # Instance details (Eq k, Num k) => Coalgebra k SSymF Source # Instance details Methodscounit :: Vect k SSymF -> k Source # (Eq k, Num k) => Algebra k SSymF Source # Instance details Methodsunit :: k -> Vect k SSymF Source #mult :: Vect k (Tensor SSymF SSymF) -> Vect k SSymF Source # (Eq k, Num k) => HasPairing k SSymF SSymF Source # A pairing showing that SSym is self-adjoint Instance details Methodspairing :: Vect k (Tensor SSymF SSymF) -> Vect k () Source # (Eq k, Num k) => HasPairing k SSymF (Dual SSymF) Source # Instance details Methodspairing :: Vect k (Tensor SSymF (Dual SSymF)) -> Vect k () Source # (Eq k, Num k) => HopfAlgebra k (Dual SSymF) Source # Instance details Methodsantipode :: Vect k (Dual SSymF) -> Vect k (Dual SSymF) Source # (Eq k, Num k) => Bialgebra k (Dual SSymF) Source # Instance details (Eq k, Num k) => Coalgebra k (Dual SSymF) Source # Instance details Methodscounit :: Vect k (Dual SSymF) -> k Source #comult :: Vect k (Dual SSymF) -> Vect k (Tensor (Dual SSymF) (Dual SSymF)) Source # (Eq k, Num k) => Algebra k (Dual SSymF) Source # Instance details Methodsunit :: k -> Vect k (Dual SSymF) Source #mult :: Vect k (Tensor (Dual SSymF) (Dual SSymF)) -> Vect k (Dual SSymF) Source #

ssymF :: [Int] -> Vect Q SSymF Source #

Construct a fundamental basis element in SSym. The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].

prop_Associative :: Eq a => (a -> a -> a) -> (a, a, a) -> Bool Source #

flatten :: (Num a1, Enum a1, Ord a2) => [a2] -> [a1] Source #

newtype SSymM Source #

An alternative "monomial" basis for the Malvenuto-Reutenauer Hopf algebra of permutations, SSym. This basis is related to the fundamental basis by Mobius inversion in the poset of permutations with the weak order.

Constructors

 SSymM [Int]
Instances
 Source # Instance details Methods(==) :: SSymM -> SSymM -> Bool #(/=) :: SSymM -> SSymM -> Bool # Source # Instance details Methods(<) :: SSymM -> SSymM -> Bool #(<=) :: SSymM -> SSymM -> Bool #(>) :: SSymM -> SSymM -> Bool #(>=) :: SSymM -> SSymM -> Bool #max :: SSymM -> SSymM -> SSymM #min :: SSymM -> SSymM -> SSymM # Source # Instance details MethodsshowsPrec :: Int -> SSymM -> ShowS #show :: SSymM -> String #showList :: [SSymM] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k SSymM Source # Instance details Methods (Eq k, Num k) => Bialgebra k SSymM Source # Instance details (Eq k, Num k) => Coalgebra k SSymM Source # Instance details Methodscounit :: Vect k SSymM -> k Source # (Eq k, Num k) => Algebra k SSymM Source # Instance details Methodsunit :: k -> Vect k SSymM Source #mult :: Vect k (Tensor SSymM SSymM) -> Vect k SSymM Source #

ssymM :: [Int] -> Vect Q SSymM Source #

Construct a monomial basis element in SSym. The list of ints must be a permutation of [1..n], eg [1,2], [3,4,2,1].

inversions :: (Num b, Enum b, Ord a) => [a] -> [(b, b)] Source #

weakOrder :: (Ord a1, Ord a2) => [a1] -> [a2] -> Bool Source #

mu :: (Num p, Eq t) => ([t], t -> t -> Bool) -> t -> t -> p Source #

ssymMtoF :: (Eq k, Num k) => Vect k SSymM -> Vect k SSymF Source #

Convert an element of SSym represented in the monomial basis to the fundamental basis

ssymFtoM :: (Eq k, Num k) => Vect k SSymF -> Vect k SSymM Source #

Convert an element of SSym represented in the fundamental basis to the monomial basis

ssymFtoDual :: (Eq k, Num k) => Vect k SSymF -> Vect k (Dual SSymF) Source #

The isomorphism from SSym to its dual that takes a permutation in the fundamental basis to its inverse in the dual basis

data PBT a Source #

A type for (rooted) planar binary trees. The basis elements of the Loday-Ronco Hopf algebra are indexed by these.

Although the trees are labelled, we're really only interested in the shapes of the trees, and hence in the type PBT (). The Algebra, Coalgebra and HopfAlgebra instances all ignore the labels. However, it is convenient to allow labels, as they can be useful for seeing what is going on, and they also make it possible to define various ways to create trees from lists of labels.

Constructors

 T (PBT a) a (PBT a) E
Instances
 Source # Instance details Methodsfmap :: (a -> b) -> PBT a -> PBT b #(<$) :: a -> PBT b -> PBT a # Eq a => Eq (PBT a) Source # Instance details Methods(==) :: PBT a -> PBT a -> Bool #(/=) :: PBT a -> PBT a -> Bool # Ord a => Ord (PBT a) Source # Instance details Methodscompare :: PBT a -> PBT a -> Ordering #(<) :: PBT a -> PBT a -> Bool #(<=) :: PBT a -> PBT a -> Bool #(>) :: PBT a -> PBT a -> Bool #(>=) :: PBT a -> PBT a -> Bool #max :: PBT a -> PBT a -> PBT a #min :: PBT a -> PBT a -> PBT a # Show a => Show (PBT a) Source # Instance details MethodsshowsPrec :: Int -> PBT a -> ShowS #show :: PBT a -> String #showList :: [PBT a] -> ShowS # newtype YSymF a Source # The fundamental basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym. Constructors  YSymF (PBT a) Instances  Source # Instance details Methodsfmap :: (a -> b) -> YSymF a -> YSymF b #(<$) :: a -> YSymF b -> YSymF a # (Eq k, Num k, Ord a) => HopfAlgebra k (YSymF a) Source # Instance details Methodsantipode :: Vect k (YSymF a) -> Vect k (YSymF a) Source # (Eq k, Num k, Ord a) => Bialgebra k (YSymF a) Source # Instance details (Eq k, Num k, Ord a) => Coalgebra k (YSymF a) Source # Instance details Methodscounit :: Vect k (YSymF a) -> k Source #comult :: Vect k (YSymF a) -> Vect k (Tensor (YSymF a) (YSymF a)) Source # (Eq k, Num k, Ord a) => Algebra k (YSymF a) Source # Instance details Methodsunit :: k -> Vect k (YSymF a) Source #mult :: Vect k (Tensor (YSymF a) (YSymF a)) -> Vect k (YSymF a) Source # Eq a => Eq (YSymF a) Source # Instance details Methods(==) :: YSymF a -> YSymF a -> Bool #(/=) :: YSymF a -> YSymF a -> Bool # Ord a => Ord (YSymF a) Source # Instance details Methodscompare :: YSymF a -> YSymF a -> Ordering #(<) :: YSymF a -> YSymF a -> Bool #(<=) :: YSymF a -> YSymF a -> Bool #(>) :: YSymF a -> YSymF a -> Bool #(>=) :: YSymF a -> YSymF a -> Bool #max :: YSymF a -> YSymF a -> YSymF a #min :: YSymF a -> YSymF a -> YSymF a # Show a => Show (YSymF a) Source # Instance details MethodsshowsPrec :: Int -> YSymF a -> ShowS #show :: YSymF a -> String #showList :: [YSymF a] -> ShowS # Graded (YSymF a) Source # Instance details Methodsgrade :: YSymF a -> Int Source #

ysymF :: PBT a -> Vect Q (YSymF a) Source #

Construct the element of YSym in the fundamental basis indexed by the given tree

nodecount :: Num p => PBT a -> p Source #

leafcount :: Num p => PBT a -> p Source #

prefix :: PBT a -> [a] Source #

shapeSignature :: Num a1 => PBT a2 -> [a1] Source #

nodeCountTree :: Num a1 => PBT a2 -> PBT a1 Source #

leafCountTree :: Num a1 => PBT a2 -> PBT a1 Source #

lrCountTree :: Num b => PBT a -> PBT (b, b) Source #

shape :: PBT a -> PBT () Source #

numbered :: Num a1 => PBT a2 -> PBT a1 Source #

splits :: PBT a -> [(PBT a, PBT a)] Source #

multisplits :: (Eq t, Num t) => t -> PBT a -> [[PBT a]] Source #

graft :: [PBT a] -> PBT a -> PBT a Source #

newtype YSymM Source #

An alternative "monomial" basis for (the dual of) the Loday-Ronco Hopf algebra of binary trees, YSym.

Constructors

 YSymM (PBT ())
Instances
 Source # Instance details Methods(==) :: YSymM -> YSymM -> Bool #(/=) :: YSymM -> YSymM -> Bool # Source # Instance details Methods(<) :: YSymM -> YSymM -> Bool #(<=) :: YSymM -> YSymM -> Bool #(>) :: YSymM -> YSymM -> Bool #(>=) :: YSymM -> YSymM -> Bool #max :: YSymM -> YSymM -> YSymM #min :: YSymM -> YSymM -> YSymM # Source # Instance details MethodsshowsPrec :: Int -> YSymM -> ShowS #show :: YSymM -> String #showList :: [YSymM] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k YSymM Source # Instance details Methods (Eq k, Num k) => Bialgebra k YSymM Source # Instance details (Eq k, Num k) => Coalgebra k YSymM Source # Instance details Methodscounit :: Vect k YSymM -> k Source # (Eq k, Num k) => Algebra k YSymM Source # Instance details Methodsunit :: k -> Vect k YSymM Source #mult :: Vect k (Tensor YSymM YSymM) -> Vect k YSymM Source #

ysymM :: PBT () -> Vect Q YSymM Source #

Construct the element of YSym in the monomial basis indexed by the given tree

trees :: Int -> [PBT ()] Source #

List all trees with the given number of nodes

tamariCovers :: PBT a -> [PBT a] Source #

The covering relation for the Tamari partial order on binary trees

tamariUpSet :: Ord a => PBT a -> [PBT a] Source #

The up-set of a binary tree in the Tamari partial order

tamariOrder :: PBT a -> PBT a -> Bool Source #

The Tamari partial order on binary trees. This is only defined between trees of the same size (number of nodes). The result between trees of different sizes is undefined (we don't check).

ysymMtoF :: (Eq k, Num k) => Vect k YSymM -> Vect k (YSymF ()) Source #

Convert an element of YSym represented in the monomial basis to the fundamental basis

ysymFtoM :: (Eq k, Num k) => Vect k (YSymF ()) -> Vect k YSymM Source #

Convert an element of YSym represented in the fundamental basis to the monomial basis

compositions :: Int -> [[Int]] Source #

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]]

quasiShuffles :: [Int] -> [Int] -> [[Int]] Source #

newtype QSymM Source #

A type for the monomial basis for the quasi-symmetric functions, indexed by compositions.

Constructors

 QSymM [Int]
Instances
 Source # Instance details Methods(==) :: QSymM -> QSymM -> Bool #(/=) :: QSymM -> QSymM -> Bool # Source # Instance details Methods(<) :: QSymM -> QSymM -> Bool #(<=) :: QSymM -> QSymM -> Bool #(>) :: QSymM -> QSymM -> Bool #(>=) :: QSymM -> QSymM -> Bool #max :: QSymM -> QSymM -> QSymM #min :: QSymM -> QSymM -> QSymM # Source # Instance details MethodsshowsPrec :: Int -> QSymM -> ShowS #show :: QSymM -> String #showList :: [QSymM] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k QSymM Source # Instance details Methods (Eq k, Num k) => Bialgebra k QSymM Source # Instance details (Eq k, Num k) => Coalgebra k QSymM Source # Instance details Methodscounit :: Vect k QSymM -> k Source # (Eq k, Num k) => Algebra k QSymM Source # Instance details Methodsunit :: k -> Vect k QSymM Source #mult :: Vect k (Tensor QSymM QSymM) -> Vect k QSymM Source # (Eq k, Num k) => HasPairing k NSym QSymM Source # A duality pairing between NSym and QSymM (monomial basis), showing that NSym and QSym are dual. Instance details Methodspairing :: Vect k (Tensor NSym QSymM) -> Vect k () Source #

qsymM :: [Int] -> Vect Q QSymM Source #

Construct the element of QSym in the monomial basis indexed by the given composition

coarsenings :: Num a => [a] -> [[a]] Source #

refinements :: [Int] -> [[Int]] Source #

newtype QSymF Source #

A type for the fundamental basis for the quasi-symmetric functions, indexed by compositions.

Constructors

 QSymF [Int]
Instances
 Source # Instance details Methods(==) :: QSymF -> QSymF -> Bool #(/=) :: QSymF -> QSymF -> Bool # Source # Instance details Methods(<) :: QSymF -> QSymF -> Bool #(<=) :: QSymF -> QSymF -> Bool #(>) :: QSymF -> QSymF -> Bool #(>=) :: QSymF -> QSymF -> Bool #max :: QSymF -> QSymF -> QSymF #min :: QSymF -> QSymF -> QSymF # Source # Instance details MethodsshowsPrec :: Int -> QSymF -> ShowS #show :: QSymF -> String #showList :: [QSymF] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k QSymF Source # Instance details Methods (Eq k, Num k) => Bialgebra k QSymF Source # Instance details (Eq k, Num k) => Coalgebra k QSymF Source # Instance details Methodscounit :: Vect k QSymF -> k Source # (Eq k, Num k) => Algebra k QSymF Source # Instance details Methodsunit :: k -> Vect k QSymF Source #mult :: Vect k (Tensor QSymF QSymF) -> Vect k QSymF Source #

qsymF :: [Int] -> Vect Q QSymF Source #

Construct the element of QSym in the fundamental basis indexed by the given composition

qsymMtoF :: (Eq k, Num k) => Vect k QSymM -> Vect k QSymF Source #

Convert an element of QSym represented in the monomial basis to the fundamental basis

qsymFtoM :: (Eq k, Num k) => Vect k QSymF -> Vect k QSymM Source #

Convert an element of QSym represented in the fundamental basis to the monomial basis

qsymPoly :: Int -> [Int] -> GlexPoly Q String Source #

qsymPoly n is is the quasi-symmetric polynomial in n variables for the indices is. (This corresponds to the monomial basis for QSym.) For example, qsymPoly 3 [2,1] == x1^2*x2+x1^2*x3+x2^2*x3.

newtype SymM Source #

A type for the monomial basis for Sym, the Hopf algebra of symmetric functions, indexed by integer partitions

Constructors

 SymM [Int]
Instances
 Source # Instance details Methods(==) :: SymM -> SymM -> Bool #(/=) :: SymM -> SymM -> Bool # Source # Instance details Methodscompare :: SymM -> SymM -> Ordering #(<) :: SymM -> SymM -> Bool #(<=) :: SymM -> SymM -> Bool #(>) :: SymM -> SymM -> Bool #(>=) :: SymM -> SymM -> Bool #max :: SymM -> SymM -> SymM #min :: SymM -> SymM -> SymM # Source # Instance details MethodsshowsPrec :: Int -> SymM -> ShowS #show :: SymM -> String #showList :: [SymM] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k SymM Source # Instance details Methods (Eq k, Num k) => Bialgebra k SymM Source # Instance details (Eq k, Num k) => Coalgebra k SymM Source # Instance details Methodscounit :: Vect k SymM -> k Source # (Eq k, Num k) => Algebra k SymM Source # Instance details Methodsunit :: k -> Vect k SymM Source #mult :: Vect k (Tensor SymM SymM) -> Vect k SymM Source # (Eq k, Num k) => HasPairing k SymH SymM Source # A duality pairing between the complete and monomial bases of Sym, showing that Sym is self-dual. Instance details Methodspairing :: Vect k (Tensor SymH SymM) -> Vect k () Source #

symM :: [Int] -> Vect Q SymM Source #

Construct the element of Sym in the monomial basis indexed by the given integer partition

compositionsFromPartition :: Eq a => [a] -> [[a]] Source #

symMult :: [Int] -> [Int] -> [[Int]] Source #

newtype SymE Source #

The elementary basis for Sym, the Hopf algebra of symmetric functions. Defined informally as > symE [n] = symM (replicate n 1) > symE lambda = product [symE [p] | p <- lambda]

Constructors

 SymE [Int]
Instances
 Source # Instance details Methods(==) :: SymE -> SymE -> Bool #(/=) :: SymE -> SymE -> Bool # Source # Instance details Methodscompare :: SymE -> SymE -> Ordering #(<) :: SymE -> SymE -> Bool #(<=) :: SymE -> SymE -> Bool #(>) :: SymE -> SymE -> Bool #(>=) :: SymE -> SymE -> Bool #max :: SymE -> SymE -> SymE #min :: SymE -> SymE -> SymE # Source # Instance details MethodsshowsPrec :: Int -> SymE -> ShowS #show :: SymE -> String #showList :: [SymE] -> ShowS # Source # Instance details Methods (Eq k, Num k) => Bialgebra k SymE Source # Instance details (Eq k, Num k) => Coalgebra k SymE Source # Instance details Methodscounit :: Vect k SymE -> k Source # (Eq k, Num k) => Algebra k SymE Source # Instance details Methodsunit :: k -> Vect k SymE Source #mult :: Vect k (Tensor SymE SymE) -> Vect k SymE Source #

symEtoM :: (Eq k, Num k) => Vect k SymE -> Vect k SymM Source #

Convert from the elementary to the monomial basis of Sym

newtype SymH Source #

The complete basis for Sym, the Hopf algebra of symmetric functions. Defined informally as > symH [n] = sum [symM lambda | lambda <- integerPartitions n] -- == all monomials of weight n > symH lambda = product [symH [p] | p <- lambda]

Constructors

 SymH [Int]
Instances
 Source # Instance details Methods(==) :: SymH -> SymH -> Bool #(/=) :: SymH -> SymH -> Bool # Source # Instance details Methodscompare :: SymH -> SymH -> Ordering #(<) :: SymH -> SymH -> Bool #(<=) :: SymH -> SymH -> Bool #(>) :: SymH -> SymH -> Bool #(>=) :: SymH -> SymH -> Bool #max :: SymH -> SymH -> SymH #min :: SymH -> SymH -> SymH # Source # Instance details MethodsshowsPrec :: Int -> SymH -> ShowS #show :: SymH -> String #showList :: [SymH] -> ShowS # (Eq k, Num k) => Bialgebra k SymH Source # Instance details (Eq k, Num k) => Coalgebra k SymH Source # Instance details Methodscounit :: Vect k SymH -> k Source # (Eq k, Num k) => Algebra k SymH Source # Instance details Methodsunit :: k -> Vect k SymH Source #mult :: Vect k (Tensor SymH SymH) -> Vect k SymH Source # (Eq k, Num k) => HasPairing k SymH SymM Source # A duality pairing between the complete and monomial bases of Sym, showing that Sym is self-dual. Instance details Methodspairing :: Vect k (Tensor SymH SymM) -> Vect k () Source #

symHtoM :: (Eq k, Num k) => Vect k SymH -> Vect k SymM Source #

Convert from the complete to the monomial basis of Sym

newtype NSym Source #

A basis for NSym, the Hopf algebra of non-commutative symmetric functions, indexed by compositions

Constructors

 NSym [Int]
Instances
 Source # Instance details Methods(==) :: NSym -> NSym -> Bool #(/=) :: NSym -> NSym -> Bool # Source # Instance details Methodscompare :: NSym -> NSym -> Ordering #(<) :: NSym -> NSym -> Bool #(<=) :: NSym -> NSym -> Bool #(>) :: NSym -> NSym -> Bool #(>=) :: NSym -> NSym -> Bool #max :: NSym -> NSym -> NSym #min :: NSym -> NSym -> NSym # Source # Instance details MethodsshowsPrec :: Int -> NSym -> ShowS #show :: NSym -> String #showList :: [NSym] -> ShowS # Source # Instance details Methods (Eq k, Num k) => HopfAlgebra k NSym Source # Instance details Methods (Eq k, Num k) => Bialgebra k NSym Source # Instance details (Eq k, Num k) => Coalgebra k NSym Source # Instance details Methodscounit :: Vect k NSym -> k Source # (Eq k, Num k) => Algebra k NSym Source # Instance details Methodsunit :: k -> Vect k NSym Source #mult :: Vect k (Tensor NSym NSym) -> Vect k NSym Source # (Eq k, Num k) => HasPairing k NSym QSymM Source # A duality pairing between NSym and QSymM (monomial basis), showing that NSym and QSym are dual. Instance details Methodspairing :: Vect k (Tensor NSym QSymM) -> Vect k () Source #

descendingTree :: Ord a => [a] -> PBT a Source #

descendingTreeMap :: (Eq k, Num k) => Vect k SSymF -> Vect k (YSymF ()) Source #

Given a permutation p of [1..n], we can construct a tree (the descending tree of p) as follows:

• Split the permutation as p = ls ++ [n] ++ rs
• 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
• To bottom out the recursion, the descending tree of the empty permutation is of course the empty tree

This map between bases SSymF -> YSymF turns out to induce a morphism of Hopf algebras.

minPerm :: Num a1 => PBT a2 -> [a1] Source #

maxPerm :: Num a1 => PBT a2 -> [a1] Source #

leftLeafCompositionMap :: (Eq k, Num k) => Vect k (YSymF a) -> Vect k QSymF Source #

A Hopf algebra morphism from YSymF to QSymF

descents :: Ord a => [a] -> [Int] Source #

descentComposition :: (Ord a1, Num a2) => [a1] -> [a2] Source #

descentMap :: (Eq k, Num k) => Vect k SSymF -> Vect k QSymF Source #

Given a permutation of [1..n], its descents are those positions where the next number is less than the previous number. 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 the permutation sequence into segments - 235-16-4 - and by counting the lengths of the segments, we get a composition 3+2+1. This map between bases SSymF -> QSymF turns out to induce a morphism of Hopf algebras.

under :: PBT a -> PBT a -> PBT a Source #

ysymmToSh :: Functor f => f YSymM -> f (Shuffle (PBT ())) Source #

symToQSymM :: (Eq k, Num k) => Vect k SymM -> Vect k QSymM Source #

The injection of Sym into QSym (defined over the monomial basis)

nsymToSymH :: (Eq k, Num k) => Vect k NSym -> Vect k SymH Source #

A surjection of NSym onto Sym (defined over the complete basis)

nsymToSSym :: (Eq k, Num k) => Vect k NSym -> Vect k SSymF Source #