Partitions
NumberOfPartitions(n) : RngIntElt -> RngIntElt
Partitions(n) : RngIntElt -> [ [ RngIntElt ] ]
Partitions(n, k) : RngIntElt, RngIntElt -> [ [ RngIntElt ] ]
RestrictedPartitions(n, M) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
RestrictedPartitions(n, k, M) : RngIntElt, SetEnum -> [ [ RngIntElt ] ]
IsPartition(S) : SeqEnum -> BoolElt
RandomPartition(n) : RngIntElt -> SeqEnum
Weight(P) : SeqEnum -> RngIntElt
IndexOfPartition(P) : SeqEnum -> RngIntElt
Example Tableau_Partitions (H145E1)
Example Tableau_RestrictedPartitions (H145E2)
Ordered Monoids
OrderedMonoid(n) : RngIntElt -> MonOrd
OrderedIntegerMonoid() : -> MonOrd
Id(O) : MonOrd -> MonOrdElt
O . i : MonOrd, RngIntElt -> MonOrdElt
O ! [w1, ..., wn] : MonOrd, [MonOrdElt] -> MonOrdElt
O ! [i1, ..., in] : MonOrd, [RngIntElt] -> MonOrdElt
w1 eq w2 : MonOrdElt, MonOrdElt -> BoolElt
w1 * w2 : MonOrdElt, MonOrdElt -> MonOrdElt
IsKnuthEquivalent(w1, w2) : MonOrdElt, MonOrdElt -> BoolElt
w[i] : MonOrdElt, RngIntElt -> RngElt
ElementToSequence(w) : MonOrdElt -> SeqEnum
Length(w) : MonOrdElt -> RngIntElt
Content(w) : MonOrdElt -> SeqEnum[RngIntElt]
IsReverseLatticeWord(w) : MonOrdElt -> BoolElt
MaximalIncreasingSequence(w) : MonOrdElt -> RngIntElt
MaximalIncreasingSequences(w, k) : SeqEnum,RngIntElt -> RngIntElt
Example Tableau_OrderedMonoid-Creation (H145E3)
Example Tableau_orderedmon--fingen (H145E4)
Example Tableau_OrderedMonoid-longest (H145E5)
Plactic Monoids
PlacticMonoid(O) : MonOrd -> MonOrd
PlacticIntegerMonoid() : -> MonOrd
OrderedMonoid(P) : MonPlc -> MonOrd
Id(P) : MonPlc -> MonPlcElt
P . i : MonPlc, RngIntElt -> MonPlcElt
P ! [u1, ..., un] : MonPlc, [MonPlcElt] -> MonPlcElt
P ! [i1, ..., in] : MonPlc, [RngIntElt] -> MonPlcElt
P ! w : MonPlc, MonOrdElt -> MonPlcElt
P ! [w1, ..., wn] : MonPlc, [MonOrdElt] -> MonPlcElt
P ! t : MonPlc, Tbl -> MonPlcElt
u1 eq u2 : MonPlcElt, MonPlcElt -> BoolElt
u1 * u2 : MonPlcElt, MonPlcElt -> MonPlcElt
Length(u) : MonPlcElt -> RngIntElt
Content(u) : MonPlcElt -> SeqEnum[RngIntElt]
Example Tableau_OrderedMonoid-basic (H145E6)
Example Tableau_OrderedMonoid-fingen (H145E7)
Tableau Monoids
TableauMonoid(O) : MonOrd -> MonTbl
TableauIntegerMonoid() : -> MonTbl
OrderedMonoid(M) : MonPlc -> MonOrd;
Example Tableau_TabMonoid-standard (H145E8)
Example Tableau_TabMonoid-fingen (H145E9)
Creation of Tableaux
Tableau(Q) : SeqEnum[RngIntElt/2] -> Tbl
Tableau(Q) : SeqEnum[MonOrdElt] -> Tbl
Tableau(S, Q) : SeqEnum[RngIntElt], SeqEnum[RngIntElt/2] -> Tbl
Tableau(S, Q) : SeqEnum[RngIntElt], SeqEnum[MonOrdElt] -> Tbl
WordToTableau(w) : MonOrdElt -> Tbl
M ! w : MonTbl, MonOrdElt -> Tbl
M ! u : MonTbl, MonPlcElt -> Tbl
M ! [i1, ..., in] : MonPlc, [MonPlcElt] -> MonPlcElt
M ! Q : MonPlc, SeqEnum -> MonPlcElt
M ! Q : MonPlc, SeqEnum -> MonPlcElt
Example Tableau_Tabcreate-basic (H145E10)
Example Tableau_Tabcreate-fingen (H145E11)
Example Tableau_Tabcreate-bang (H145E12)
Enumeration of Tableaux
StandardTableaux(P) : SeqEnum[RngIntElt] -> SetEnum
StandardTableauxOfWeight(n) : RngIntElt -> SetEnum
TableauxOfShape(S, m) : SeqEnum[RngIntElt], RngIntElt -> SetEnum
TableauxOnShapeWithContent(S, C) : SeqEnum[RngIntElt], SeqEnum[RngIntElt] -> SetEnum
TableauxWithContent(C) : SeqEnum[RngIntElt] -> SetEnum
Example Tableau_MultTabCreate1 (H145E13)
Example Tableau_MultTabcreate-fingen (H145E14)
Random Tableaux
RandomHookWalk(P, i, j) : SeqEnum[RngIntElt], RngIntElt, RngIntElt -> RngIntElt, RngIntElt
RandomTableau(S) : SeqEnum[RngIntElt] -> Tbl
RandomTableau(n) : RngIntElt -> Tbl
Example Tableau_Tab-Random (H145E15)
Basic Access Functions
Shape(t) : Tbl -> SeqEnum[RngIntElt]
SkewShape(t) : Tbl -> SeqEnum[RngIntElt]
PartitionCovers(P1, P2) : SeqEnum, SeqEnum -> BoolElt
ConjugatePartition(P) : SeqEnum -> SeqEnum
Weight(t) : Tbl -> RngIntElt
SkewWeight(t) : Tbl -> RngIntElt
NumberOfRows(t) : Tbl -> RngIntElt
NumberOfSkewRows(t) : Tbl -> RngIntElt
Row(t, i) : Tbl, RngIntElt -> MonOrdElt
Rows(t) : Tbl -> SeqEnum[MonOrdElt]
Column(t, j) : Tbl, RngIntElt -> MonOrdElt
Columns(t) : Tbl -> SeqEnum[MonOrdElt]
RowSkewLength(t, i) : Tbl,RngIntElt -> RngIntElt
ColumnSkewLength(t, j) : Tbl,RngIntElt -> RngIntElt
FirstIndexOfRow(t, i) : Tbl,RngIntElt -> RngIntElt
LastIndexOfRow(t, i) : Tbl,RngIntElt -> RngIntElt
FirstIndexOfColumn(t, j) : Tbl,RngIntElt -> RngIntElt
LastIndexOfColumn(t, j) : Tbl,RngIntElt -> RngIntElt
Example Tableau_Tab-Access (H145E16)
Properties
HookLength(t, i, j) : Tbl, RngIntElt, RngIntElt -> RngIntElt
Content(t) : Tbl -> SeqEnum
Word(t) : Tbl -> MonOrdElt
ColumnWord(t) : Tbl -> SeqEnum
IsStandard(t) : Tbl -> BoolElt
IsSkew(t) : Tbl -> BoolElt
IsLittlewoodRichardson(t) : Tbl -> BoolElt
Example Tableau_Tab-check-standard (H145E17)
Example Tableau_Tab-check-words (H145E18)
Operations
t1 eq t2 : MonPlcElt, MonPlcElt -> BoolElt
t1 * t2 : Tbl, Tbl -> Tbl
DiagonalSum(t1, t2) : Tbl,Tbl -> Tbl
Conjugate(t) : Tbl -> Tbl
JeuDeTaquin(~t, i, j) : Tbl, RngIntElt, RngIntElt ->
JeuDeTaquin(~t) : Tbl ->
InverseJeuDeTaquin(~t, i, j) : Tbl, RngIntElt, RngIntElt ->
RowInsert(~t, w) : Tbl, MonOrdElt ->
RowInsert(~t, x) : Tbl, RngIntElt ->
InverseRowInsert(~t, i, j) : Tbl, RngIntElt, RngIntElt ->
Example Tableau_Tab-Jeu (H145E19)
Example Tableau_Tab-Comp-Mult (H145E20)
The Robinson-Schensted-Knuth Correspondence
LexicographicalOrdering(~w1, ~w2) : MonOrdElt, MonOrdElt ->
IsLexicographicallyOrdered(w1, w2) : MonOrdElt, MonOrdElt -> boolean
RSKCorrespondence(w) : SeqEnum[RngIntElt] -> Tbl, Tbl
InverseRSKCorrespondenceSingleWord(t1, t2) : Tbl, Tbl -> MonOrdElt
RSKCorrespondence(w1, w2) : SeqEnum[RngIntElt], SeqEnum[RngIntElt] -> Tbl,Tbl
InverseRSKCorrespondenceDoubleWord(t1, t2) : Tbl, Tbl -> MonOrdElt, MonOrdElt
RSKCorrespondence(M) : Mtrx -> Tbl, Tbl
InverseRSKCorrespondenceMatrix(t1, t2) : Tbl, Tbl -> Mat
Example Tableau_RSK-singleword (H145E21)
Example Tableau_RSK-doubleword (H145E22)
Example Tableau_RSK-Matrix (H145E23)
Example Tableau_RSK-perms (H145E24)
Counting Tableaux
NumberOfStandardTableaux(P) : SeqEnum -> RngIntElt
NumberOfStandardTableauxOnWeight(n) : RngIntElt -> RngIntElt
NumberOfTableauxOnAlphabet(P, m) : SeqEnum,RngIntElt -> RngIntElt
KostkaNumber(S, C) : SeqEnum[RngIntElt], SeqEnum[RngIntElt] -> RngIntElt
Example Tableau_CountStandardTab (H145E25)
Example Tableau_CountTabAlph-Binomial (H145E26)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012