NewTableaux

Standard and semistandard (skew) Young tableaux, RSK, promotion, evacuation, crystal operators, border strips, TeX output; semistandard augmented fillings (SSAF) with statistics, crystals and Mason insertion.

Load with Needs["NewTableaux`"].

Background on symmetricfunctions.com: Partitions, permutations and tableaux, RSK, the Robinson–Schensted–Knuth correspondence, Crystals, Operations on Young tableaux, Border strip tableaux and the Littlewood map.

Functions and symbols

ArrayToBiword

ArrayToBiword[a] converts a nonnegative integer array a to a biword listing the positions of its entries.

AtomFillings

AtomFillings[alpha] returns the augmented fillings enumerating the Demazure atom indexed by alpha.

Background: Key polynomials and Demazure atoms

BinaryMatrixToBiword

BinaryMatrixToBiword[m] converts a binary matrix m to its biword of positions of 1 entries.

BiwordRSK

BiwordRSK[{w1, w2}] applies row-insertion RSK to two equal-length words and returns a pair of YoungTableau objects.
BiwordRSK[w] applies RSK with the increasing word Range[Length[w]] as the first word.

Background: RSK, the Robinson–Schensted–Knuth correspondence

BiwordRSKDual

BiwordRSKDual[{w1, w2}] applies dual row-insertion RSK to two equal-length words and returns a pair of YoungTableau objects.
BiwordRSKDual[w] applies dual RSK with the increasing word Range[Length[w]] as the first word.

Background: RSK, the Robinson–Schensted–Knuth correspondence

BorderStrips

BorderStrips[shape,size] returns a list of pairs
{new-shape,strip} of possible border-strips to remove.

Background: Border strip tableaux and the Littlewood map

BorderStripTableaux

BorderStripTableaux[shape, type] returns a list of all border-strip tableaux of the shape.

Background: Border strip tableaux and the Littlewood map

BSTHeightVector

BSTHeightVector[bst] returns a vector where vi is the height of strip i.

ChargeToMajMap

ChargeToMajMap[ssaf] applies the charge-to-major-index map to an augmented filling.

ColumnLatticePaths

ColumnLatticePaths[ssyt] returns a graphical representation of the ssyt
as a set of non-intersecting lattice paths, each path corresponding to a column in the ssyt

CrystalEi

CrystalEi[ssyt,i] performs the crystal raising operator ei on the tableau.
It also works on lists

Background: Crystals

CrystalFi

CrystalFi[ssyt,i] performs the crystal lowering operator fi on the tableau. It also works on words.

Background: Crystals

CrystalSi

CrystalSi[ssyt,i] performs the crystal
transposition operator si on the tableau. It also works on words.

Background: Crystals

CylindricSYT

CylindricSYT[lam, k] returns all standard Young tableaux of cylindric shape lam with shift k. CylindricSYT[{lam, mu}, k] uses skew shape lam/mu; k defaults to 0.

CylindricTableaux

CylindricTableaux[{lam,mu},k]
produces all cylindric tableaux with partition weight and shifted up k steps from minimal possible shift.

HasOuterCornerQ

HasOuterCornerQ[tab] returns True when the skew shape of tab has an outer corner in the legacy OldYoungTableaux sense.
HasOuterCornerQ[{lam, mu}] applies the same test to a skew shape.

InvMajStatistic

InvMajStatistic[tab] returns the modified
Macdonald polynomial statistics {inv,maj}. Works on skew shapes.

KeyFillings

KeyFillings[alpha] returns the augmented fillings enumerating the key polynomial indexed by alpha.

Background: Key polynomials and Demazure atoms

KnuthRepresentative

KnuthRepresentative[pi] returns the unique permutation which is Knuth equivalent to pi, and is the reading word of some SYT.

LascouxSchutzenberger

LascouxSchutzenberger[object,i] applies the Lascoux–Schutzenberger involution.

LineBreaks

LineBreaks is an option for YTableauTeX; its default is True.

LongestIncreasingSubsequence

LongestIncreasingSubsequence[w] returns the length of the longest increasing subsequence.

PlanePartitions

PlanePartitions[shape,max] returns all plane partitions of given shape, with entries <= max.
PlanePartitions[a,b,c] returns all a x b plane partitions with entries bounded by c.

RowLatticePaths

RowLatticePaths[ssyt] returns a graphical representation of the ssyt
as a set of non-intersecting lattice paths, each path corresponding to a row in the ssyt

RPPToAtom

RPPToAtom[rpp] applies Mason’s column-set insertion map to an augmented reverse plane partition.

SemiStandardYoungTableaux

SemiStandardYoungTableaux[{lam,mu},w] returns
a list of all SSYT with given shape and weight.

SpecialRimHookTableaux

SpecialRimHookTableaux[shape,type] returns all special rim-hook tableaux of given shape and type.

SSAF

SSAF[rows] represents a semistandard augmented filling with basement entries in the first column.

SSAFBasement

SSAFBasement[ssaf] returns the basement entries of ssaf.

SSAFCoInversions

SSAFCoInversions[ssaf] returns the number of coinversion triples of ssaf.

SSAFColumnSets

SSAFColumnSets[ssaf, start] returns sorted column sets, starting at column start (default 2).

SSAFCrystalString

SSAFCrystalString[ssaf,i] returns the i-crystal string containing ssaf.

Background: Crystals

SSAFCrystalWord

SSAFCrystalWord[ssaf,i] returns the uncancelled i-crystal word of ssaf.

Background: Crystals

SSAFDn

SSAFDn[ssaf] returns the number of unequal horizontal adjacencies of ssaf.

SSAFForm

SSAFForm[ssaf, options] returns a graphical representation of an augmented filling.

SSAFillings

SSAFillings[alpha, basement] returns all valid augmented fillings of shape alpha and basement.

SSAFInversions

SSAFInversions[ssaf] returns the number of inversion triples of ssaf.

SSAFKnownCharge

SSAFKnownCharge[ssaf] returns the charge when the augmented filling has partition shape.

SSAFMajorIndex

SSAFMajorIndex[ssaf] returns the augmented filling major index.

SSAFMonomial

SSAFMonomial[ssaf,x] returns the weight monomial of ssaf.

SSAFQ

SSAFQ[ssaf] returns True if ssaf is a valid semistandard augmented filling.

SSAFShape

SSAFShape[ssaf] returns the weak composition of non-basement row lengths.

SSAFWeight

SSAFWeight[ssaf] returns the weight vector of the non-basement entries of ssaf.

SSAFWeightNormalize

SSAFWeightNormalize[ssaf] applies crystal involutions until the weight is a partition.

SSYTCharge

SSYTCharge[ssyt] returns the charge of a semistandard Young tableau.

SSYTCocharge

SSYTCocharge[ssyt] returns the cocharge of a semistandard Young tableau.

SSYTKPromotion

SSYTKPromotion[ssyt,k] performs k-promotion.

Background: Operations on Young tableaux

SSYTKPromotionInverse

SSYTKPromotionInverse[ssyt,k] performs the inverse of k-promotion. This also works on skew shapes!

Background: Operations on Young tableaux

SSYTToAtom

SSYTToAtom[tab] applies Mason’s insertion map from a semistandard tableau to an atom filling.

Background: Key polynomials and Demazure atoms

StandardYoungTableaux

StandardYoungTableaux[n],
StandardYoungTableaux[lam] or StandardYoungTableaux[{lam,mu}] returns a list of SYTs

SuperStandardTableau

SuperStandardTableau[{lam,mu}] returns the SYT with 1,2,.. in first row and so on.

SYTCharge

SYTCharge[ssyt] returns the charge of a semistandard tableau, with partition weight.

SYTCocharge

SYTCocharge[ssyt] returns the cocharge of a semistandard tableau.

SYTDescents

SYTDescents[syt] returns the number of descents of a standard Young tableau.

SYTDescentSet

SYTDescentSet[syt] returns the descent set of the standard Young tableau.

SYTDualMajorIndex

SYTDualMajorIndex[syt] returns the dual major index of a standard Young tableau.

SYTEvacuation

SYTEvacuation[syt] performs the evacuation involution on the SYT. Does not work on skew shapes or SSYTs.

Background: Operations on Young tableaux

SYTEvacuationDual

SYTEvacuationDual[syt] is the dual evacuation.

Background: Operations on Young tableaux

SYTMajorIndex

SYTMajorIndex[syt] returns the major index of a standard Young tableau.

SYTMax

SYTMax[tab] returns the maximum entry in the tableau.

SYTPromotion

SYTPromotion[syt,[k]] performes the promotion operator k times. Default is 1 time.

Background: Operations on Young tableaux

SYTReadingWord

SYTReadingWord[tab] returns the reading word of tab, formed by reading rows from bottom to top and omitting None entries.

SYTSize

SYTSize[tab] returns the number of boxes in the tableau.

SYTStandardize

SYTStandardize[ssyt] standardizes the (skew) ssyt.

TableauShortTeX

TableauShortTeX[tab] returns the \tableaushort{..} TeX string for the tableau.

TAtomFillings

TAtomFillings[alpha] returns the non-attacking augmented fillings used by the t-atom identity.

Background: Key polynomials and Demazure atoms

UseArray

UseArray is an option for YTableauTeX; its default is True and False selects the legacy \young representation.

YoungDiagramForm

YoungDiagramForm[lam, options] returns a graphical representation of the Young diagram of partition lam. YoungDiagramForm[{lam, mu}, options] uses skew shape lam/mu. ItemSize defaults to 1 and DescentSet defaults to False.

YoungTableau

YoungTableau[data] represents a Young tableau.

YoungTableauForm

YoungTableauForm[tab, options] returns a graphical representation of the Young tableau. ItemSize defaults to 1 and DescentSet defaults to False.

YoungTableauShape

YoungTableauShape[tab] returns the outer shape of the tableau.
YoungTableauShape[tab,m] returns the shape of formed by all entries <=m.

YoungTableauSize

YoungTableauSize[tab] returns the number of boxes in the tableau, excluding skew boxes represented by None.

YoungTableauWeight

YoungTableauWeight[tab] returns the weight vector counting entries 1 through the maximum entry of tab.

YTableauTeX

YTableauTeX[tab, options] returns a TeX string for tab. LineBreaks defaults to True and UseArray selects the legacy \young representation when False.