QuasiSymmetricFunctions

Monomial, fundamental and power-sum quasisymmetric functions, quasisymmetric Schur functions, and bridges to polynomials and to symmetric functions.

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Background on symmetricfunctions.com: Quasisymmetric functions, Gessel quasisymmetric functions, Quasisymmetric Schur functions and variants.

Functions and symbols

FundamentalQSymbol

FundamentalQSymbol[alpha, x] represents the fundamental quasisymmetric-function basis element indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Gessel quasisymmetric functions

FundamentalQSymmetric

FundamentalQSymmetric[alpha, x] returns the fundamental quasisymmetric function indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Gessel quasisymmetric functions

MonomialQSymbol

MonomialQSymbol[alpha, x] represents the monomial quasisymmetric-function basis element indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

MonomialQSymmetric

MonomialQSymmetric[alpha, x] returns the monomial quasisymmetric function indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

PartitionedCompositionCoarsenings

PartitionedCompositionCoarsenings[alpha] returns all coarsenings of the partitioned composition alpha, represented as a list of lists of compositions.

Background: Quasisymmetric functions

PolynomialToQuasiSymmetricFunction

PolynomialToQuasiSymmetricFunction[poly, x, basisSymbol] converts a quasisymmetric polynomial in x[1], x[2], … to the MonomialQSymbol or FundamentalQSymbol basis; the basis defaults to MonomialQSymbol.

Background: Quasisymmetric functions

PowerSumAltQSymmetric

PowerSumAltQSymmetric[alpha, x] returns the alternate quasisymmetric power-sum function indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

PowerSumQSymbol

PowerSumQSymbol[alpha, x] represents the quasisymmetric power-sum basis element indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

PowerSumQSymmetric

PowerSumQSymmetric[alpha, x] returns the quasisymmetric power-sum function indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

QuasiSchurQSymbol

QuasiSchurQSymbol[alpha, x] represents the HLMW quasisymmetric Schur basis element indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric Schur functions and variants

QuasiSchurQSymmetric

QuasiSchurQSymmetric[alpha, basisSymbol] returns the HLMW quasisymmetric Schur function indexed by composition alpha in the MonomialQSymbol basis, or in the FundamentalQSymbol basis when requested.

Background: Quasisymmetric Schur functions and variants

QuasiSymmetricFunctionToPolynomial

QuasiSymmetricFunctionToPolynomial[expr, x, n] expresses a quasisymmetric function in the variables x[1] through x[n].

Background: Quasisymmetric functions

ToFundamentalBasis

ToFundamentalBasis[poly, x] converts poly to the fundamental quasisymmetric basis. The alphabet x defaults to None.

Background: Quasisymmetric functions

ToOtherQSymmetricBasis

ToOtherQSymmetricBasis[basis, pol, newSymb, x, mm] converts pol from the monomial basis to the basis specified by basis and newSymb. The alphabet x defaults to None and the monomial symbol mm defaults to MonomialQSymbol.

Background: Quasisymmetric functions

ToPowerSumQSymBasis

ToPowerSumQSymBasis[poly, x] converts poly to the quasisymmetric power-sum basis. The alphabet x defaults to None.

Background: Quasisymmetric functions

ToQuasiSymmetric

ToQuasiSymmetric[expr] embeds a SymmetricFunctions expression in QSym by sending m_lambda to the sum of M_alpha over all distinct rearrangements alpha of lambda.

Background: Quasisymmetric functions

ToZPowerSumQSymBasis

ToZPowerSumQSymBasis[poly, x] converts poly to the z-normalized quasisymmetric power-sum basis. The alphabet x defaults to None.

Background: Quasisymmetric functions

ZPowerSumQSymbol

ZPowerSumQSymbol[alpha, x] represents the z-normalized quasisymmetric power-sum basis element indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions

ZPowerSumQSymmetric

ZPowerSumQSymmetric[alpha, x] returns the z-normalized quasisymmetric power-sum function indexed by composition alpha in alphabet x. The alphabet x defaults to None.

Background: Quasisymmetric functions