ACE, an environment for
algebraic combinatorics
AceBanner - print a banner about ACE
AceBuildData - build and store data
AceLoadData - load data from disk
AceSysInfo - give information on the system
AceTime - give time information
AceUnloadData - unload data
AceWhichIs - look for a function in a list of packages
BN and the hyperoctahedral groups
BnBar2Code - code of a signed permutation
BnBar2Mat - signed permutation matrix from a signed permutation
BnCode2Bar - signed permutation corresponding to a code
BnCode2Interval - list of codes of all group elements below a given one
BnCode2Inv - code of the inverse of a group element
BnCode2Length - length of a code
BnCode2ListRd - list of all reduced decompositions of a code
BnCode2Palin - palindromic permutation corresponding to a code
BnCode2Rd - canonical reduced decomposition of a code
BnGenCode - generate one code
BnListCode - generate all codes
BnMat2Bar - calculate the signed permutation from a given permutation matrix
BnMultBar - multiply signed permutations
BnMultCode - multiply codes
BnPalin2Code - code corresponding to a palindromic permutation
BnRd2Code - calculate a code from a decomposition into simple reflections
BnTranspoPalin - image of a simple reflection in S(2n)
Algebras related to hyperoctahedral groups: package BNA
BnGaMult - multiplication of two elements of the Bn
BnGaOnPol - action of an element of the Bn
BnGaYang - Yang-Baxter element of the Bn-group algebra
BnIdcaMult - multiplication of two elements of the Bn
BnIdcaOnPol - action of an element of the Bn
BnIdcaYang - Yang-Baxter element of the Bn-idCoxeter algebra
BnNcaMult - multiplication of two elements of the Bn
BnNcaOnPol - action of an element of the Bn
BnNcaYang - Yang-Baxter element of the Bn-nilCoxeter algebra
Bna2Table - convert an element of any Bn
BnaAdd - sum of two elements of any Bn
BnaCode2Bar - convert codes into signed permutations
BnaCode2Palin - convert codes into palindromic permutations
BnaMinus - difference of two elements of any Bn
BnaNormal - normalize an element of a Bn
BnaPalin2Code - convert palindromic permutations into codes
Rd2BnGa - calculate a product of generators
Rd2BnIdca - calculate a product of generators
Rd2BnNca - calculate a product of generators
Table2Bna - convert a table into an element of any Bn
CG: a package about classical Lie groups
CgMultiplicity - calculate weight multiplicities
CgPositiveRoots - positive roots of a classical group
CgRho - half the sum of positive roots
CgSchurSeries - truncated Schur function series
O2s - from the orthogonal Schur basis to the Schur basis
ODimRep - calculate the dimension of an irreducible representation of the orthogonal group
OJtMat - Jacobi-Trudi-type matrix
Sp2s - from the symplectic Schur basis to the Schur basis
SpDimRep - calculate the dimension of an irreducible representation of the symplectic group
SpJtMat - Jacobi-Trudi-type matrix
ToO - express any symmetric function in the basis of orthogonal Schur functions
ToSp - express any symmetric function in the basis of symplectic Schur functions
CLG, a package to handle characters of linear groups
CLG_n - fix the order of the linear group
Gauss - compute a Gaussian polynomial
GlDimRep - compute the dimension of a representation
Toe_n - convert any symmetric function to a e-polynomial
Toh_n - convert any symmetric function to a h-polynomial
Tom_n - express any symmetric function in the basis of monomial symmetric functions
Top_n - convert any symmetric function to a p-polynomial
Tos_n - express any symmetric function in the basis of Schur functions
Tox_n - express any symmetric function in the basis of monomials
x2m_n - from the basis of monomials to monomial symmetric function basis
COMP, a package to handle compositions
CompCompo - compare two compositions
Compo2Conjugate - conjugate a composition
Compo2MajorIndex - major index of a given composition
Compo2Mat - a planar representation of a composition
ListCompo - list of compositions of a given weight
ListCompoFatter - all compositions fatter than a given one
ListCompoFiner - all compositions finer that a given one
Computations in the ring of polynomials with FM
FM_n - fix the number of variables
ToXfix - convert any expression to the X Schubert basis of the ring of polynomials as a free module over Sym
ToXfixScal - convert any expression to the X Schubert basis of the ring of polynomials as a free module over Sym
X2Xfix - from X Schubert basis to X Schubert basis of the ring of polynomials as a free module over Sym
Xfix2X - from X Schubert basis with symmetric functions as coefficients to X Schubert basis
Xfix2x - from X Schubert basis with symmetric functions as coefficients to the basis of monomials
x2Xfix - from the basis of monomials to X Schubert basis of the ring of polynomials as a free module over Sym
x2XfixScal - from the basis of monomials to X Schubert basis of the ring of polynomials as a free module over Sym
FREE, a package to work in the free algebra
Free2NilFree - inverse plaxification map
Free2Plax - quotient of the free algebra by plactic relations
Free2PlaxClass - replace each word by its plactic class
Free2Pol - from words to polynomials
Free2StdFree - standardization of words
Free2Table - convert an element of the free algebra into a table
FreeAdd - sum of two elements of the free algebra
FreeConcat - concatenation product in the free algebra
FreeMinus - difference of two elements of the free algebra
FreeNormal - normalize an element of the free algebra
FreeShuffle - shuffle product in the free algebra
IdcaOnFree - action of the idCoxeter algebra on words
NilFree2Free - plaxification map
Pol2Free - from polynomials to words
Pol2Row - from polynomials to increasing words
Sf2Free - from symmetric polynomials to words
SgaOnFree - action of the symmetric group algebra on words
Table2Free - convert a table into an element of the free algebra
Working in the generic Hecke algebra with HEKA
Heka2Table - convert an element of the Hecke algebra into a table
HekaAdd - sum of two elements of the Hecke algebra
HekaCharTable - compute the table of characters
HekaJucis - Jucis-Murphy element of the Hecke algebra
HekaMinus - difference of two elements of the Hecke algebra
HekaMult - multiplication of two elements of the Hecke algebra
HekaNormal - normalize an element of the Hecke algebra
HekaOnPol - action of an element of the algebra on a polynomial
HekaOnX - action of an element of the Hecke algebra of the symmetric group on a linear
combination of X[perm]
HekaOnXfix - action of an element of the Hecke algebra of the symmetric group on the
ring of polynomials as a free module over Sym
HekaYang - compute a special element of the Hecke algebra
Rd2Heka - calculate a product of generators
Table2Heka - convert a table into an element of the Hecke algebra
IDCA to compute in the idCoxeter algebra
Idca2Demazure - Demazure element of the idCoxeter algebra
Idca2Table - convert an element of the idCoxeter algebra into a table
IdcaAdd - sum of two elements of the idCoxeter algebra
IdcaMinus - difference of two elements of the idCoxeter algebra
IdcaMult - multiplication of two elements of the idCoxeter algebra
IdcaNormal - normalize an element of the idCoxeter algebra
IdcaOnPol - action of an element of the algebra on a polynomial
IdcaOnX - action of an element of the idCoxeter algebra on a linear combination of X[perm]
IdcaOnXfix - action of an element of the idCoxeter algebra on the ring of polynomials as a free
module over Sym
IdcaYang - compute a special element of the idCoxeter algebra
Rd2Idca - calculate a product of generators
Table2Idca - convert a table into an element of the idCoxeter algebra
NCA to compute in the nilCoxeter algebra
Nca2Table - convert an element of the nilCoxeter algebra into a table
NcaAdd - sum of two elements of the nilCoxeter algebra
NcaMinus - difference of two elements of the nilCoxeter algebra
NcaMult - multiplication of two elements of the nilCoxeter algebra
NcaNormal - normalize an element of the nilCoxeter algebra
NcaOnPol - action of an element of the algebra on a polynomial
NcaOnX - action of an element of the nilCoxeter algebra on a linear combination of X[perm]
NcaOnXX - action of an element of the nilCoxeter algebra on a linear combination of
XX[perm]
NcaOnXfix - action of an element of the nilCoxeter algebra on the ring of polynomials as a free module over Sym
NcaOnY - action of an element of the nilCoxeter algebra on a linear combination of Y[code]
NcaOnYY - action of an element of the nilCoxeter algebra on a linear combination of
YY[code]
NcaYang - compute a special element of the nilCoxeter algebra
Rd2Nca - calculate a product of generators
Table2Nca - convert a table into an element of the nilCoxeter algebra
Noncommutative and quasi-symmetric functions with NCSF
NcsfInternal - internal product
NcsfMat - transition matrix from X-basis to Y-basis
NcsfPairing - computes the scalar product of quasi-symmetric function and noncommutative symmetric function
NcsfPlethysm - computes the plethysm of symmetric function and a quasi-symmetric function
NcsfTransAlpha - specialization A/(1-q) --> A or A --> A/(1-q)
Qsf2x - from quasi-symmetric function to the basis of monomials
QsfMat - transition matrix from X-basis to Y-basis
Sf2Qsf - expand a symmetric function on F basis (dual basis of noncommutative ribbon Schur function)
ToE - express any quasi-symmetric function in the E-basis (dual basis of noncommutative
elementary symmetric function)
ToF - express any quasi-symmetric function in the F-basis (dual basis of Ribbon Schur basis)
ToL - express any noncommutative symmetric function in the L-basis (noncommutative elementary
symmetric function)
ToM - express any quasi-symmetric function in the M-basis (quasi-monomial symmetric functions)
ToPh - express any noncommutative symmetric function in the Ph-basis (noncommutative power
sums symmetric function of the second kind)
ToPs - express any noncommutative symmetric function in the Ps-basis (noncommutative power
sums symmetric function of the first kind)
ToQPh - express any quasi-symmetric function in the QPh-basis (dual basis of noncommutative
power sums symmetric function of the second kind)
ToQPs - express any quasi-symmetric function in the QPs-basis (dual basis of noncommutative
power sums symmetric functions of the first kind)
ToR - express any noncommutative symmetric function in the R-basis (ribbon Schur function)
ToS - express any noncommutative symmetric function in the S-basis (noncommutative complete
symmetric function)
PART, a package to handle partitions
Border2Part - partition associated to a given border
CompPart - compare two partitions
Diagonal2Part - partition from a list of diagonals
Exp2Part - partition from its exponential notation
Frob2Part - from Frobenius notation to standard notation for partitions
ListPart - list of partitions of a given weight
ListPartIn - list of partitions contained into a given partition or between two given
partitions
ListSkewDiag - list of skew diagrams
PCore2Part - partition from p-core and p-quotient
Part2Border - border of a given partition
Part2Conjugate - conjugate a partition
Part2Diagonal - diagonals of the given partition
Part2Exp - multiplicities of a given partition
Part2Frob - from standard partition notation to Frobenius notation
Part2ListHook - calculate the list of hook lengths
Part2Mat - a planar representation of a given partition
Part2PCore - p-core and p-quotient of a partition
PartOrderMat - order matrix on partitions
SkewPart2Mat - a planar representation of a skew partition
SplitPart - split a partition into several ones
SFA to handle symmetric functions on different alphabets
Pol2SfA - from the basis of monomials to SFA
Sf2SfA - from SYMF objects to SFA objects
SfA2Sf - from SFA objects to SYMF objects
SfA2TableVar - symmetric functions appearing in an expression
SfACollect - collect products that are on algebraic bases
SfAExpand - expand a SFA expression
SfAOmega - apply the omega-automorphism
SfAVars - specify valid variables in alphabets
ToeA - develop any symmetric function on the e-basis
TohA - develop any symmetric function on the h-basis
TomA - develop any symmetric function on the m-basis
TopA - develop any symmetric function on the p-basis
TosA - develop any symmetric function on the s-basis
SG, a package for the symmetric group
Code2Perm - calculate a permutation from its code
CompPerm - compare two permutations
Cycle2Perm - calculate the permutation from its decomposition into disjoint cycles
Desc2ListPerm - generate all permutations with a given descent set
FactorPerm - test whether a permutation factorizes into a direct product
GenPerm - generate an element of the symmetric group
ListPerm - generate all permutations
Mat2Perm - calculate a permutation from its matrix of permutation
MultPerm - multiply permutations
Part2SgCC - compute a conjugacy class of the symmetric group
Perm2Betti - compute the Betti polynomial
Perm2Code - calculate the code of a permutation
Perm2Cycle - decompose a permutation into disjoint cycles
Perm2CycleType - compute the cycle type of a permutation
Perm2Desc - compute the list of descents of a permutation
Perm2Interval - compute all codes of permutations below a given one
Perm2Inv - invert a permutation
Perm2Length - calculate the length of a permutation
Perm2ListInv - calculate the set of inversions of a permutation
Perm2ListRd - list all reduced decompositions of a given permutation
Perm2Mat - calculate the permutation matrix
Perm2RRep - calculate the matrix representing a permutation in the regular representation
Perm2Rd - calculate a canonical reduced decomposition of a permutation
Perm2Rise - compute the list of rises of a permutation
Perm2Rothe - calculate the Rothe diagram of a permutation
Perm2Sign - return the signature of a permutation
RRep2Perm - calculate a permutation from its matrix in the regular representation
RandPerm - return a random permutation
Rd2Perm - calculate a permutation from a decomposition
SgCharTable - compute the table of characters
SgDimRep - compute the dimension of a representation
SgTranspo - return a transposition
Vect2Perm - compute a canonical permutation from a vector
Using SGA to work with the symmetric group algebra
Part2SgaCIdemp - compute a central idempotent of the symmetric group algebra
Rd2Sga - calculate a product of generators
Sga2Carre - Carre element of the symmetric group algebra
Sga2Nabla - Nabla element of the symmetric group algebra
Sga2Table - convert an element of the symmetric group algebra into a table
SgaAdd - sum of two elements of the symmetric group algebra
SgaJucis - Jucis-Murphy element of the symmetric group algebra
SgaMinus - difference of two elements of the symmetric group algebra
SgaMult - multiplication of two elements of the symmetric group algebra
SgaNormal - reduce each permutation inside an expression to the minimum symmetric group
containing it
SgaOnPol - action of an element of the algebra on an expression
SgaOnX - action of an element of the symmetric group algebra on a linear combination of X[perm]
SgaOnXfix - action of an element of the symmetric group algebra on the ring of polynomials as a free module over Sym
SgaYang - compute a special element of the symmetric group algebra
Table2Sga - convert a table into an element of the symmetric group algebra
Manipulating Schubert polynomials with SP
Flag - fix the degree of the symmetric group
NewtonInterp - Newton Interpolation formula
Sp2TableVar - variable set for the SP package
SpScalarPol - scalar product on polynomials
SpSpecialize - specialize a set of variables
TableX - table of all Schubert polynomials
TableXX - table of all double Schubert polynomials
TableY - table of all Schubert polynomials
TableYY - table of all double Schubert polynomials
ToX - convert any expression to the X Schubert basis
ToXX - convert any expression to the XX Schubert basis
ToY - convert any expression to the Y Schubert basis
ToYY - convert any expression to the YY Schubert basis
Tox - express any expression in the basis of monomials
X2Dim - specialization in X Schubert basis
x2X - from the basis of monomials to X Schubert basis
x2XX - from the basis of monomials to XX Schubert basis
SYMF to handle symmetric functions
Char2Sf - convert a virtual character into a symmetric function
Sf2Char - convert a symmetric function into a virtual character
Sf2Table - convert a symmetric function into a table
Sf2TableVar - variable set of a symmetric function
SfAddBasis - addition of a new basis
SfCCProd - compute the conjugacy class product of symmetric functions
SfDiff - differential operator
SfDualBasis - addition of a new basis, dual to an existing one
SfEval - realize the action of symmetric functions on polynomials
SfInternal - compute the inner tensor product of symmetric functions
SfJtMat - compute the Jacobi-Trudi matrix
SfMat - transition matrix between symmetric functions bases
SfOmega - apply the omega-automorphism
SfPlethysm - plethysm of symmetric functions
SfScalar - scalar product of symmetric functions
SfTheta - apply the theta-automorphism
SfZee - compute the scalar product of a p-function with itself
Table2Sf - convert a table into a symmetric function
Toc - express any symmetric function in the basis of cycle indices
Toe - convert any symmetric function to a e-polynomial
Toh - convert any symmetric function to a h-polynomial
Tom - express any symmetric function in the basis of monomial symmetric functions
Top - convert any symmetric function to a p-polynomial
Tos - express any symmetric function in the basis of Schur functions
Using TAB to manipulate tableaux
Bump - insert a word into a tableau
BumpC - insert a word into a contretableau
CTab2Mat - convert a contretableau into a matrix
CTab2Part - compute the shape of a contretableau
CTab2Word - convert a contretableau into a word
InvSchensted - inverse Schensted correspondence
ListStdTab - generate all standard tableaux
ListTab - generate all tableaux of a given shape and evaluation
ListYama - generate all Yamanouchi words of a given evaluation
Mat2CTab - convert a matrix into a contretableau
Mat2Tab - convert a matrix into a tableau
Part2ListStdTab - generate all standard tableaux of a given shape
PermOnCTab - action of a permutation on a product of columns
PermOnTab - action of a permutation on a product of rows
Schensted - Schensted correspondence
StdTab2SgaIdemp - build an idempotent from a standard tableau
StdTab2Yama - convert a standard tableau into a Yamanouchi word
Tab2Mat - convert a tableau into a matrix
Tab2Part - compute the shape of a tableau
Tab2Word - convert a tableau into a word
Word2CTab - convert a word into a contretableau
Word2Charge - returns the charge of a word
Word2CoCharge - returns the cocharge of a word
Word2Key - left or right key
Word2Tab - convert a word into a tableau
Yama2StdTab - convert a Yamanouchi word into a standard tableau
A package for checking types
IsA - test whether the argument is a basis element of
an algebra related to the symmetric group
IsB - test whether the argument is a basis element of
an algebra related to the hyperoctahedral group
IsBnBar - test whether the argument is a signed permutation
IsBorder - test whether the argument is a border of a partition
IsCTab - test whether the argument is a contretableau
IsBnCode - test whether the argument is a code
IsCompo - test whether the argument is a composition
IsDiagonal - test whether the argument is a diagonal
coding of a partition
IsO - test whether the argument is an element of the
orthogonal Schur basis
IsBnPalin - test whether the argument is a
palindromic permutation
IsPart - test whether the argument is a partition
IsPe - test whether the argument is an element of the Pe-basis
IsPerm - test whether the argument is a permutation
IsRegPart - test whether the argument is a n-regular partition
IsSkewDiag - test whether the argument is a skew diagram
IsSkewPart - test whether the argument is a skew partition
IsSp - test whether the argument is an element of the
symplectic Schur basis
IsStdCTab - test whether the argument is a
standard contretableau
IsStdTab - test whether the argument is a
standard tableau
IsTab - test whether the argument is a tableau
IsWord - test whether the argument is a word
IsX - test whether the argument is a Schubert polynomial
IsXX - test whether the argument is a double Schubert polynomial
IsY - test whether the argument is a Schubert polynomial
IsYY - test whether the argument is a Schubert polynomial
IsYama - test whether the argument is a Yamanouchi word
Isx - test whether the argument is an x.i indeterminate
Isy - test whether the argument is an y.i indeterminate
Isc - test whether the argument is an element of the
cycle index basis
Iscl - test whether the argument is like cl[part]
Ise - test whether the argument is a generator of the e-basis
IseA - test whether the argument is a product of elements
of the e-basis on a formal alphabet expression
Ish - test whether the argument is a generator of the h-basis
IshA - test whether the argument is a product of elements
of the h-basis on a formal alphabet expression
Ism - test whether the argument is an element of the
monomial basis
IsmA - test whether the argument is an element of the
monomial basis on a formal alphabet expression
Isp - test whether the argument is a generator of the p-basis
IspA - test whether the argument is a product of elements
of the p-basis on a formal alphabet expression
Iss - test whether the argument is an element of the
Schur basis
IssA - test whether the argument is an element of the
Schur basis on a formal alphabet expression