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Tableaux -- a package for computing with Young tableaux

Description

This package provides the classes SkewTableau and YoungTableau.

Author

Version

This documentation describes version 0.5 of Tableaux, released July 22, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{TableauxSource,
  title = {{Tableaux: constructing skew tableaux. Version~0.5}},
  author = {John Graf},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

  • Types
    • SkewTableau -- a type of HashTable representing a skew Young tableau
    • YoungTableau -- a type of HashTable representing a (nonskew) Young tableau
  • Functions and commands
  • Methods
    • allSemistandardTableaux(Partition) -- see allSemistandardTableaux -- list all semistandard Young tableaux of a given shape
    • allSemistandardTableaux(Partition,Partition) -- see allSemistandardTableaux -- list all semistandard Young tableaux of a given shape
    • allSemistandardTableaux(Partition,Partition,ZZ) -- see allSemistandardTableaux -- list all semistandard Young tableaux of a given shape
    • allSemistandardTableaux(Partition,ZZ) -- see allSemistandardTableaux -- list all semistandard Young tableaux of a given shape
    • applyEntries(SkewTableau,Function) -- see applyEntries -- apply a function to all entries in the tableau
    • applyPositions(SkewTableau,Function) -- see applyPositions -- apply a function to all positions of boxes in a tableau
    • boxContent(ZZ,ZZ) -- see boxContent -- compute the content of a box of a tableau
    • columnEntries(SkewTableau,ZZ) -- see columnEntries -- get the entries in a column.
    • columnEntries(ZZ,SkewTableau) -- see columnEntries -- get the entries in a column.
    • columnRange(SkewTableau) -- see columnRange -- the range of column indices of a tableau.
    • components(SkewTableau) -- get the connected components of a tableau
    • conjugate(SkewTableau) -- conjugate a tableau
    • entries(SkewTableau) -- the filling of a skew tableau
    • ferrersDiagram(Partition) -- see ferrersDiagram -- a net of the Ferrers diagram
    • ferrersDiagram(Partition,Partition) -- see ferrersDiagram -- a net of the Ferrers diagram
    • ferrersDiagram(SkewTableau) -- see ferrersDiagram -- a net of the Ferrers diagram
    • hookLength(Sequence,SkewTableau) -- see hookLength -- compute the hook length of a box of a tableau
    • isNonnegative(Partition) -- see isNonnegative -- check if shapes are nonnegative
    • isNonnegative(Partition,Partition) -- see isNonnegative -- check if shapes are nonnegative
    • isNonnegative(SkewTableau) -- see isNonnegative -- check if shapes are nonnegative
    • isWeaklyDecreasing(Partition) -- see isWeaklyDecreasing -- check if shapes are weakly decreasing
    • isWeaklyDecreasing(Partition,Partition) -- see isWeaklyDecreasing -- check if shapes are weakly decreasing
    • isWeaklyDecreasing(SkewTableau) -- see isWeaklyDecreasing -- check if shapes are weakly decreasing
    • numColumns(SkewTableau) -- the number of columns in a tableau
    • numRows(SkewTableau) -- the number of rows in a tableau
    • numSemistandardTableaux(Partition) -- see numSemistandardTableaux -- compute the number of semistandard Young tableaux of a given shape
    • numSemistandardTableaux(Partition,ZZ) -- see numSemistandardTableaux -- compute the number of semistandard Young tableaux of a given shape
    • positionList(SkewTableau) -- see positionList -- get the positions of all the boxes in a tableau
    • rowEntries(SkewTableau,ZZ) -- see rowEntries -- get the entries in a row.
    • rowEntries(ZZ,SkewTableau) -- see rowEntries -- get the entries in a row.
    • rowRange(SkewTableau) -- see rowRange -- the range of row indices of a tableau.
    • shape(YoungTableau) -- see shape -- the shape of a Young tableau
    • truncate(Partition) -- see shape -- the shape of a Young tableau
    • shift(SkewTableau) -- see shift -- shift a tableau
    • shift(SkewTableau,ZZ) -- see shift -- shift a tableau
    • size(SkewTableau) -- the number of boxes in a skew tableau
    • pad(Partition,Partition) -- see skewShape -- the shape of a skew tableau
    • skewShape(SkewTableau) -- see skewShape -- the shape of a skew tableau
    • standardize(Partition,Partition) -- see skewShape -- the shape of a skew tableau
    • truncate(Partition,Partition) -- see skewShape -- the shape of a skew tableau
    • net(SkewTableau) -- see SkewTableau -- a type of HashTable representing a skew Young tableau
    • drawInnerShape(Boolean) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(Partition) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(Partition,List) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(Partition,Partition) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(Partition,Partition,List) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(Sequence,List) -- see skewTableau -- constructor for type SkewTableau
    • skewTableau(YoungTableau) -- see skewTableau -- constructor for type SkewTableau
    • SkewTableau ++ SkewTableau -- direct sum of tableaux
    • SkewTableau ^ ZZ -- get the entries in a row.
    • SkewTableau _ Sequence -- get the entry at a specific position.
    • SkewTableau _ ZZ -- get the entries in a column.
    • SkewTableau || SkewTableau -- vertical concatenation of two tableaux
    • tex(SkewTableau) -- LaTeX output for a skew tableau
    • toIndex(Sequence,SkewTableau) -- see toIndex -- get the index of a box, given its position.
    • toIndex(SkewTableau,Sequence) -- see toIndex -- get the index of a box, given its position.
    • toPosition(SkewTableau,ZZ) -- see toPosition -- get the position of a box, given its index.
    • toPosition(ZZ,SkewTableau) -- see toPosition -- get the position of a box, given its index.
    • unshift(SkewTableau) -- see unshift -- unshift a tableau
    • unshift(SkewTableau,ZZ) -- see unshift -- unshift a tableau
    • verticalConcatenate(List) -- see verticalConcatenate -- vertical concatenation of a list of tableaux
    • youngDiagram(Partition) -- see youngDiagram -- a net of the Young diagram
    • youngDiagram(Partition,Partition) -- see youngDiagram -- a net of the Young diagram
    • youngDiagram(SkewTableau) -- see youngDiagram -- a net of the Young diagram
    • youngTableau(Partition) -- see youngTableau -- constructor for type YoungTableau
    • youngTableau(Partition,List) -- see youngTableau -- constructor for type YoungTableau

For the programmer

The object Tableaux is a package, defined in Tableaux.m2, with auxiliary files in Tableaux/.

The source of this document is in Tableaux/documentation.m2:12:0.