TableauTuples

A TableauTuple is a tuple of tableaux. These objects arise naturally in representation theory of the wreath products of cyclic groups and the symmetric groups where the standard tableau tuples index bases for the ordinary irreducible representations. This generalises the well-known fact the ordinary irreducible representations of the symmetric groups have bases indexed by the standard tableaux of a given shape. More generally, TableauTuples, or multitableaux, appear in the representation theory of the degenerate and non-degenerate cyclotomic Hecke algebras and in the crystal theory of the integral highest weight representations of the affine special linear groups.

A TableauTuple is an ordered tuple \((t^{(1)}, t^{(2)}, \ldots, t^{(l)})\) of tableaux. The length of the tuple is its level and the tableaux \(t^{(1)}, t^{(2)}, \ldots, t^{(l)}\) are the components of the TableauTuple.

A tableaux can be thought of as the labelled diagram of a partition. Analogously, a TableauTuple is the labelled diagram of a PartitionTuple. That is, a TableauTuple is a tableau of PartitionTuple shape. As much as possible, TableauTuples behave in exactly the same way as Tableaux. There are obvious differences in that the cells of a partition are ordered pairs \((r, c)\), where \(r\) is a row index and \(c\) a column index, whereas the cells of a PartitionTuple are ordered triples \((k, r, c)\), with \(r\) and \(c\) as before and \(k\) indexes the component.

Frequently, we will call a TableauTuple a tableau, or a tableau of PartitionTuple shape. If the shape of the tableau is known this should not cause any confusion.

Warning

In sage the convention is that the \((k, r, c)\)-th entry of a tableau tuple \(t\) is the entry in row \(r\), column \(c\) and component \(k\) of the tableau. This is because it makes much more sense to let t[k] be component of the tableau. In particular, we want t(k,r,c) == t[k][r][c]. In the literature, the cells of a tableau tuple are usually written in the form \((r, c, k)\), where \(r\) is the row index, \(c\) is the column index, and \(k\) is the component index.

The same convention applies to the cells of PartitionTuples.

Note

As with partitions and tableaux, the cells are 0-based. For example, the (lexicographically) first cell in any non-empty tableau tuple is [0,0,0].

EXAMPLES:

sage: TableauTuple([[1,2,3],[4,5]])
[[1, 2, 3], [4, 5]]
sage: t = TableauTuple([ [[6,7],[8,9]],[[1,2,3],[4,5]] ]); t
([[6, 7], [8, 9]], [[1, 2, 3], [4, 5]])
sage: t.pp()
  6  7     1  2  3
  8  9     4  5
sage: t(0,0,1)
7
sage: t(1,0,1)
2
sage: t.shape()
([2, 2], [3, 2])
sage: t.size()
9
sage: t.level()
2
sage: t.components()
[[[6, 7], [8, 9]], [[1, 2, 3], [4, 5]]]
sage: t.entries()
[6, 7, 8, 9, 1, 2, 3, 4, 5]
sage: t.parent()
Tableau tuples
sage: t.category()
Category of elements of Tableau tuples

One reason for implementing TableauTuples is to be able to consider StandardTableauTuples. These objects arise in many areas of algebraic combinatorics. In particular, they index bases for the Specht modules of the cyclotomic Hecke algebras of type \(G(r,1,n)\). A StandardTableauTuple of tableau whose entries are increasing along rows and down columns in each component and which contain the numbers \(1,2, \ldots, n\), where the shape of the StandardTableauTuple is a PartitionTuple of \(n\).

sage: s = StandardTableauTuple([ [[1,2],[3]],[[4,5]]])
sage: s.category()
Category of elements of Standard tableau tuples
sage: t = TableauTuple([ [[1,2],[3]],[[4,5]]])
sage: t.is_standard(), t.is_column_strict(), t.is_row_strict()
(True, True, True)
sage: t.category()
Category of elements of Tableau tuples
sage: s == t
True
sage: s is t
False
sage: s == StandardTableauTuple(t)
True
sage: StandardTableauTuples([ [2,1],[1] ])[:]
[([[1, 2], [3]], [[4]]),
 ([[1, 3], [2]], [[4]]),
 ([[1, 2], [4]], [[3]]),
 ([[1, 3], [4]], [[2]]),
 ([[2, 3], [4]], [[1]]),
 ([[1, 4], [2]], [[3]]),
 ([[1, 4], [3]], [[2]]),
 ([[2, 4], [3]], [[1]])]

As tableaux (of partition shape) are in natural bijection with 1-tuples of tableaux all of the TableauTuple classes return an ordinary Tableau when given TableauTuple of level 1.

sage: TableauTuples( level=1 ) is Tableaux()
True
sage: TableauTuple([[1,2,3],[4,5]])
[[1, 2, 3], [4, 5]]
sage: TableauTuple([ [[1,2,3],[4,5]] ])
[[1, 2, 3], [4, 5]]
sage: TableauTuple([[1,2,3],[4,5]]) == Tableau([[1,2,3],[4,5]])
True

There is one situation where a 1-tuple of tableau is not actually a Tableau; tableaux generated by the StandardTableauTuples() iterators must have the correct parents, so in this one case 1-tuples of tableaux are different from Tableaux:

sage: StandardTableauTuples()[:10]
[(),
 ([[1]]),
 ([], []),
 ([[1, 2]]),
 ([[1], [2]]),
 ([[1]], []),
 ([], [[1]]),
 ([], [], []),
 ([[1, 2, 3]]),
 ([[1, 3], [2]])]

AUTHORS:

• Andrew Mathas (2012-10-09): Initial version – heavily based on tableau.py by Mike Hansen (2007) and Jason Bandlow (2011).

• Andrew Mathas (2016-08-11): Row standard tableaux added

Element classes:

• TableauTuples

• StandardTableauTuples

• RowStandardTableauTuples

Factory classes:

• TableauTuples

• StandardTableauTuples

• RowStandardTableauTuples

Parent classes:

• TableauTuples_all

• TableauTuples_level

• TableauTuples_size

• TableauTuples_level_size

• StandardTableauTuples_all

• StandardTableauTuples_level

• StandardTableauTuples_size

• StandardTableauTuples_level_size

• StandardTableauTuples_shape

• StandardTableaux_residue

• StandardTableaux_residue_shape

• RowStandardTableauTuples_all

• RowStandardTableauTuples_level

• RowStandardTableauTuples_size

• RowStandardTableauTuples_level_size

• RowStandardTableauTuples_shape

• RowStandardTableauTuples_residue

• RowStandardTableauTuples_residue_shape

See also

• Tableau

• StandardTableau

• Tableaux

• StandardTableaux

• Partitions

• PartitionTuples

• ResidueSequence

Todo

Implement semistandard tableau tuples as defined in [DJM1998].

Much of the combinatorics implemented here is motivated by this and subsequent papers on the representation theory of these algebras.

class sage.combinat.tableau_tuple.RowStandardTableauTuple(parent, t, check=True)

Bases: sage.combinat.tableau_tuple.TableauTuple

A class for row standard tableau tuples of shape a partition tuple.

A row standard tableau tuple of size \(n\) is an ordered tuple of row standard tableaux (see RowStandardTableau), with entries \(1, 2, \ldots, n\) such that, in each component, the entries are in increasing order along each row. If the tableau in component \(k\) has shape \(\lambda^{(k)}\) then \(\lambda=(\lambda^{(1)},\ldots,\lambda^{(l)}\) is a PartitionTuple.

Note

The tableaux appearing in a RowStandardTableauTuple are row strict, but individually they are not standard tableaux because the entries in any single component of a RowStandardTableauTuple will typically not be in bijection with \(\{1, 2, \ldots, n\}\).

INPUT:

• t – a tableau, a list of (standard) tableau or an equivalent list

OUTPUT:

• A RowStandardTableauTuple object constructed from t.

Note

Sage uses the English convention for (tuples of) partitions and tableaux: the longer rows are displayed on top. As with PartitionTuple, in sage the cells, or nodes, of partition tuples are 0-based. For example, the (lexicographically) first cell in any non-empty partition tuple is \([0,0,0]\). Further, the coordinates [k,r,c] in a TableauTuple refer to the component, row and column indices, respectively.

EXAMPLES:

sage: t = RowStandardTableauTuple([[[4,7],[3]],[[2,6,8],[1,5]],[[9]]]); t
([[4, 7], [3]], [[2, 6, 8], [1, 5]], [[9]])
sage: t.pp()
  4  7     2  6  8     9
  3        1  5
sage: t.shape()
([2, 1], [3, 2], [1])
sage: t[0].pp()  # pretty printing
  4  7
  3
sage: t.is_row_strict()
True
sage: t[0].is_standard()
False
sage: RowStandardTableauTuple([[],[],[]]) # An empty tableau tuple
([], [], [])
sage: RowStandardTableauTuple([[[4,5],[6]],[[1,2,3]]]) in StandardTableauTuples()
True
sage: RowStandardTableauTuple([[[5,6],[4]],[[1,2,3]]]) in StandardTableauTuples()
False

When using code that will generate a lot of tableaux, it is slightly more efficient to construct a RowStandardTableauTuple from the appropriate parent object:

sage: RST = RowStandardTableauTuples()
sage: RST([[[4,5],[7]],[[1,2,3],[6,8]],[[9]]])
([[4, 5], [7]], [[1, 2, 3], [6, 8]], [[9]])

See also

• RowTableau

• RowTableaux

• TableauTuples

• TableauTuple

• StandardTableauTuples

• StandardTableauTuple

• RowStandardTableauTuples

codegree(e, multicharge)

Return the Brundan-Kleshchev-Wang [BKW2011] codegree of self.

The codegree of a tableau is an integer that is defined recursively by successively stripping off the number \(k\), for \(k = n, n-1, \ldots, 1\) and at stage adding the number of addable cell of the same residue minus the number of removable cells of the same residue as \(k\) and which are above \(k\) in the diagram.

The codegree of the tableau self gives the degree of “dual” homogeneous basis element of the graded Specht module which is indexed by self.

INPUT:

• e – the quantum characteristic

• multicharge – the multicharge

OUTPUT:

The codegree of the tableau self, which is an integer.

EXAMPLES:

sage: StandardTableauTuple([[[1]], [], []]).codegree(0,(0,0,0))
0
sage: StandardTableauTuple([[],[[1]], []]).codegree(0,(0,0,0))
1
sage: StandardTableauTuple([[], [], [[1]]]).codegree(0,(0,0,0))
2
sage: StandardTableauTuple([[[1]],[[2]], []]).codegree(0,(0,0,0))
-1
sage: StandardTableauTuple([[[1]], [], [[2]]]).codegree(0,(0,0,0))
0
sage: StandardTableauTuple([[],[[1]], [[2]]]).codegree(0,(0,0,0))
1
sage: StandardTableauTuple([[[2]],[[1]], []]).codegree(0,(0,0,0))
1
sage: StandardTableauTuple([[[2]], [], [[1]]]).codegree(0,(0,0,0))
2
sage: StandardTableauTuple([[],[[2]], [[1]]]).codegree(0,(0,0,0))
3

degree(e, multicharge)

Return the Brundan-Kleshchev-Wang [BKW2011] degree of self.

The degree of a tableau is an integer that is defined recursively by successively stripping off the number \(k\), for \(k = n, n-1, \ldots, 1\), and at stage adding the count of the number of addable cell of the same residue minus the number of removable cells of them same residue as \(k\) and that are below \(k\) in the diagram.

Note that even though this degree function was defined by Brundan-Kleshchev-Wang [BKW2011] the underlying combinatorics is much older, going back at least to Misra and Miwa.

The degrees of the tableau \(T\) gives the degree of the homogeneous basis element of the graded Specht module which is indexed by \(T\).

INPUT:

• e – the quantum characteristic e

• multicharge – (default: [0]) the multicharge

OUTPUT:

The degree of the tableau self, which is an integer.

EXAMPLES:

sage: StandardTableauTuple([[[1]], [], []]).degree(0,(0,0,0))
2
sage: StandardTableauTuple([[],[[1]], []]).degree(0,(0,0,0))
1
sage: StandardTableauTuple([[], [], [[1]]]).degree(0,(0,0,0))
0
sage: StandardTableauTuple([[[1]],[[2]], []]).degree(0,(0,0,0))
3
sage: StandardTableauTuple([[[1]], [], [[2]]]).degree(0,(0,0,0))
2
sage: StandardTableauTuple([[],[[1]], [[2]]]).degree(0,(0,0,0))
1
sage: StandardTableauTuple([[[2]],[[1]], []]).degree(0,(0,0,0))
1
sage: StandardTableauTuple([[[2]], [], [[1]]]).degree(0,(0,0,0))
0
sage: StandardTableauTuple([[],[[2]], [[1]]]).degree(0,(0,0,0))
-1

inverse(k)

Return the cell containing k in the tableau tuple self.

EXAMPLES:

sage: RowStandardTableauTuple([[[3,4],[1,2]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(1)
(0, 1, 0)
sage: RowStandardTableauTuple([[[3,4],[1,2]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(2)
(0, 1, 1)
sage: RowStandardTableauTuple([[[3,4],[1,2]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(3)
(0, 0, 0)
sage: RowStandardTableauTuple([[[3,4],[1,2]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(4)
(0, 0, 1)
sage: StandardTableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(1)
(0, 0, 0)
sage: StandardTableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(2)
(0, 0, 1)
sage: StandardTableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(3)
(0, 1, 0)
sage: StandardTableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).inverse(12)
(2, 2, 0)

residue_sequence(e, multicharge)

Return the sage.combinat.tableau_residues.ResidueSequence of self.

INPUT:

• e – integer in \(\{0, 2, 3, 4, 5, \ldots\}\)

• multicharge – a sequence of integers of length equal to the level/length of self

OUTPUT:

The residue sequence of the tableau.

EXAMPLES:

sage: RowStandardTableauTuple([[[5]],[[3,4],[1,2]]]).residue_sequence(3,[0,0])
3-residue sequence (2,0,0,1,0) with multicharge (0,0)
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue_sequence(3,[0,1])
3-residue sequence (1,2,0,1,0) with multicharge (0,1)
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue_sequence(3,[0,2])
3-residue sequence (2,0,1,2,0) with multicharge (0,2)

class sage.combinat.tableau_tuple.RowStandardTableauTuples

Bases: sage.combinat.tableau_tuple.TableauTuples

A factory class for the various classes of tuples of row standard tableau.

INPUT:

There are three optional arguments:

• level – the level() of the tuples of tableaux

• size – the size() of the tuples of tableaux

• shape – a list or a partition tuple specifying the shape() of the row standard tableau tuples

It is not necessary to use the keywords. If they are not used then the first integer argument specifies the level() and the second the size() of the tableau tuples.

OUTPUT:

The appropriate subclass of RowStandardTableauTuples.

A tuple of row standard tableau is a tableau whose entries are positive integers which increase from left to right along the rows in each component. The entries do NOT need to increase from left to right along the components.

Note

Sage uses the English convention for (tuples of) partitions and tableaux: the longer rows are displayed on top. As with PartitionTuple, in sage the cells, or nodes, of partition tuples are 0-based. For example, the (lexicographically) first cell in any non-empty partition tuple is \([0,0,0]\).

EXAMPLES:

sage: tabs = RowStandardTableauTuples([[2],[1,1]]); tabs
Row standard tableau tuples of shape ([2], [1, 1])
sage: tabs.cardinality()
12
sage: tabs[:]
[([[3, 4]], [[2], [1]]),
 ([[2, 4]], [[3], [1]]),
 ([[1, 4]], [[3], [2]]),
 ([[1, 2]], [[4], [3]]),
 ([[1, 3]], [[4], [2]]),
 ([[2, 3]], [[4], [1]]),
 ([[1, 4]], [[2], [3]]),
 ([[1, 3]], [[2], [4]]),
 ([[1, 2]], [[3], [4]]),
 ([[2, 3]], [[1], [4]]),
 ([[2, 4]], [[1], [3]]),
 ([[3, 4]], [[1], [2]])]

sage: tabs = RowStandardTableauTuples(level=3); tabs
Row standard tableau tuples of level 3
sage: tabs[100]
([], [], [[2, 3], [1]])

sage: RowStandardTableauTuples()[0]
([])

See also

• TableauTuples

• Tableau

• RowStandardTableau

• RowStandardTableauTuples

Element

alias of RowStandardTableauTuple

level_one_parent_class

alias of sage.combinat.tableau.RowStandardTableaux_all

shape()

Return the shape of the set of RowStandardTableauTuples, or None if it is not defined.

EXAMPLES:

sage: tabs=RowStandardTableauTuples(shape=[[5,2],[3,2],[],[1,1,1],[3]]); tabs
Row standard tableau tuples of shape ([5, 2], [3, 2], [], [1, 1, 1], [3])
sage: tabs.shape()
([5, 2], [3, 2], [], [1, 1, 1], [3])
sage: RowStandardTableauTuples().shape() is None
True

class sage.combinat.tableau_tuple.RowStandardTableauTuples_all

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Default class of all RowStandardTableauTuples with an arbitrary level() and size().

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: RowStandardTableauTuples().an_element()
([[4, 5, 6, 7]], [[2, 3]], [[1]])

class sage.combinat.tableau_tuple.RowStandardTableauTuples_level(level)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all RowStandardTableauTuples with a fixed level and arbitrary size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: RowStandardTableauTuples(2).an_element()
([[1]], [[2, 3]])
sage: RowStandardTableauTuples(3).an_element()
([[1]], [[2, 3]], [[4, 5, 6, 7]])

class sage.combinat.tableau_tuple.RowStandardTableauTuples_level_size(level, size)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all RowStandardTableauTuples with a fixed level and a fixed size.

an_element()

Returns a particular element of self.

EXAMPLES:

sage: RowStandardTableauTuples(5,size=2).an_element()
([], [], [], [], [[1], [2]])
sage: RowStandardTableauTuples(2,size=4).an_element()
([[1]], [[2, 3], [4]])

class sage.combinat.tableau_tuple.RowStandardTableauTuples_residue(residue)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples

Class of all row standard tableau tuples with a fixed residue sequence.

Implicitly, this also specifies the quantum characteristic, multicharge and hence the level and size of the tableaux.

Note

This class is not intended to be called directly, but rather, it is accessed through the row standard tableaux.

EXAMPLES:

sage: RowStandardTableau([[3,4,5],[1,2]]).residue_sequence(2).row_standard_tableaux()
Row standard tableaux with 2-residue sequence (1,0,0,1,0) and multicharge (0)
sage: RowStandardTableau([[3,4,5],[1,2]]).residue_sequence(3).row_standard_tableaux()
Row standard tableaux with 3-residue sequence (2,0,0,1,2) and multicharge (0)
sage: RowStandardTableauTuple([[[5,6],[7]],[[1,2,3],[4]]]).residue_sequence(2,(0,0)).row_standard_tableaux()
Row standard tableaux with 2-residue sequence (0,1,0,1,0,1,1) and multicharge (0,0)
sage: RowStandardTableauTuple([[[5,6],[7]],[[1,2,3],[4]]]).residue_sequence(3,(0,1)).row_standard_tableaux()
Row standard tableaux with 3-residue sequence (1,2,0,0,0,1,2) and multicharge (0,1)

an_element()

Return a particular element of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3).row_standard_tableaux().an_element()
[[2, 3], [1]]
sage: StandardTableau([[1,3],[2]]).residue_sequence(3).row_standard_tableaux().an_element()
[[1, 3], [2]]
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().an_element()
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().an_element()
([[4], [3], [1], [2]], [])

level()

Return the level of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3,(0,1)).row_standard_tableaux().level()
2
sage: StandardTableau([[1,2],[3]]).residue_sequence(3,(0,1)).row_standard_tableaux().level()
2
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().level()
2
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().level()
2

multicharge()

Return the multicharge of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3,(0,1)).row_standard_tableaux().multicharge()
(0, 1)
sage: StandardTableau([[1,2],[3]]).residue_sequence(3,(0,1)).row_standard_tableaux().multicharge()
(0, 1)
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().multicharge()
(0, 1)
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().multicharge()
(0, 1)

quantum_characteristic()

Return the quantum characteristic of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3,(0,1)).row_standard_tableaux().quantum_characteristic()
3
sage: StandardTableau([[1,2],[3]]).residue_sequence(3,(0,1)).row_standard_tableaux().quantum_characteristic()
3
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().quantum_characteristic()
3
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().quantum_characteristic()
3

residue_sequence()

Return the residue sequence of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3,(0,1)).row_standard_tableaux().residue_sequence()
3-residue sequence (2,0,1) with multicharge (0,1)
sage: StandardTableau([[1,2],[3]]).residue_sequence(3,(0,1)).row_standard_tableaux().residue_sequence()
3-residue sequence (0,1,2) with multicharge (0,1)
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().residue_sequence()
3-residue sequence (0,1,2,0) with multicharge (0,1)
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().residue_sequence()
3-residue sequence (1,0,2,0) with multicharge (0,1)

size()

Return the size of self.

EXAMPLES:

sage: RowStandardTableau([[2,3],[1]]).residue_sequence(3,(0,1)).row_standard_tableaux().size()
3
sage: StandardTableau([[1,2],[3]]).residue_sequence(3,(0,1)).row_standard_tableaux().size()
3
sage: RowStandardTableauTuple([[[4]],[[2,3],[1]]]).residue_sequence(3,(0,1)).row_standard_tableaux().size()
4
sage: StandardTableauTuple([[[4]],[[1,3],[2]]]).residue_sequence(3,(0,1)).row_standard_tableaux().size()
4

class sage.combinat.tableau_tuple.RowStandardTableauTuples_residue_shape(residue, shape)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples_residue

All row standard tableau tuples with a fixed residue and shape.

INPUT:

• shape – the shape of the partitions or partition tuples

• residue – the residue sequence of the label

EXAMPLES:

sage: res = RowStandardTableauTuple([[[3,6],[1]],[[5,7],[4],[2]]]).residue_sequence(3,(0,0))
sage: tabs = res.row_standard_tableaux([[2,1],[2,1,1]]); tabs
Row standard (2,1|2,1^2)-tableaux with 3-residue sequence (2,1,0,2,0,1,1) and multicharge (0,0)
sage: tabs.shape()
([2, 1], [2, 1, 1])
sage: tabs.level()
2
sage: tabs[:6]
[([[5, 7], [4]], [[3, 6], [1], [2]]),
 ([[5, 7], [1]], [[3, 6], [4], [2]]),
 ([[3, 7], [4]], [[5, 6], [1], [2]]),
 ([[3, 7], [1]], [[5, 6], [4], [2]]),
 ([[5, 6], [4]], [[3, 7], [1], [2]]),
 ([[5, 6], [1]], [[3, 7], [4], [2]])]

class sage.combinat.tableau_tuple.RowStandardTableauTuples_shape(shape)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples

Class of all RowStandardTableauTuples of a fixed shape.

an_element()

Returns a particular element of self.

EXAMPLES:

sage: RowStandardTableauTuples([[2],[2,1]]).an_element()
([[4, 5]], [[1, 3], [2]])
sage: RowStandardTableauTuples([[10],[],[]]).an_element()
([[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], [], [])

cardinality()

Return the number of row standard tableau tuples of with the same shape as the partition tuple self.

This is just the index of the corresponding Young subgroup in the full symmetric group.

EXAMPLES:

sage: RowStandardTableauTuples([[3,2,1],[]]).cardinality()
60
sage: RowStandardTableauTuples([[1],[1],[1]]).cardinality()
6
sage: RowStandardTableauTuples([[2,1],[1],[1]]).cardinality()
60

class sage.combinat.tableau_tuple.RowStandardTableauTuples_size(size)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all RowStandardTableauTuples with an arbitrary level and a fixed size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: RowStandardTableauTuples(size=2).an_element()
([[1]], [[2]], [], [])
sage: RowStandardTableauTuples(size=4).an_element()
([[1]], [[2, 3, 4]], [], [])

class sage.combinat.tableau_tuple.StandardTableauTuple(parent, t, check=True)

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuple

A class to model a standard tableau of shape a partition tuple. This is a tuple of standard tableau with entries \(1, 2, \ldots, n\), where \(n\) is the size of the underlying partition tuple, such that the entries increase along rows and down columns in each component of the tuple.

sage: s=StandardTableauTuple([[1,2,3],[4,5]]) sage: t=StandardTableauTuple([[1,2],[3,5],[4]]) sage: s.dominates(t) True sage: t.dominates(s) False sage: StandardTableauTuple([[1,2,3],[4,5]]) in RowStandardTableauTuples() True

The tableaux appearing in a StandardTableauTuple are both row and column strict, but individually they are not standard tableaux because the entries in any single component of a StandardTableauTuple will typically not be in bijection with \(\{1, 2, \ldots, n\}\).

INPUT:

• t – a tableau, a list of (standard) tableau or an equivalent list

OUTPUT:

• A StandardTableauTuple object constructed from t.

Note

Sage uses the English convention for (tuples of) partitions and tableaux: the longer rows are displayed on top. As with PartitionTuple, in sage the cells, or nodes, of partition tuples are 0-based. For example, the (lexicographically) first cell in any non-empty partition tuple is \([0,0,0]\). Further, the coordinates [k,r,c] in a TableauTuple refer to the component, row and column indices, respectively.

EXAMPLES:

sage: t=TableauTuple([ [[1,3,4],[7,9]], [[2,8,11],[6]], [[5,10]] ]); t
([[1, 3, 4], [7, 9]], [[2, 8, 11], [6]], [[5, 10]])
sage: t[0][0][0]
1
sage: t[1][1][0]
6
sage: t[2][0][0]
5
sage: t[2][0][1]
10

sage: t = StandardTableauTuple([[[4,5],[7]],[[1,2,3],[6,8]],[[9]]]); t
([[4, 5], [7]], [[1, 2, 3], [6, 8]], [[9]])
sage: t.pp()
  4  5     1  2  3     9
  7        6  8
sage: t.shape()
([2, 1], [3, 2], [1])
sage: t[0].pp()  # pretty printing
  4  5
  7
sage: t.is_standard()
True
sage: t[0].is_standard()
False
sage: StandardTableauTuple([[],[],[]]) # An empty tableau tuple
([], [], [])

When using code that will generate a lot of tableaux, it is slightly more efficient to construct a StandardTableauTuple from the appropriate parent object:

sage: STT = StandardTableauTuples()
sage: STT([[[4,5],[7]],[[1,2,3],[6,8]],[[9]]])
([[4, 5], [7]], [[1, 2, 3], [6, 8]], [[9]])

See also

• Tableau

• Tableaux

• TableauTuples

• TableauTuple

• StandardTableauTuples

dominates(t)

Return True if the tableau (tuple) self dominates the tableau t. The two tableaux do not need to be of the same shape.

EXAMPLES:

sage: s = StandardTableauTuple([[1,2,3],[4,5]])
sage: t = StandardTableauTuple([[1,2],[3,5],[4]])
sage: s.dominates(t)
True
sage: t.dominates(s)
False

restrict(m=None)

Returns the restriction of the standard tableau self to m, which defaults to one less than the current size().

EXAMPLES:

sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(6)
([[5]], [[1, 2], [3, 4]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(5)
([[5]], [[1, 2], [3, 4]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(4)
([], [[1, 2], [3, 4]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3)
([], [[1, 2], [3]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(2)
([], [[1, 2]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(1)
([], [[1]])
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(0)
([], [])

Where possible the restricted tableau belongs to the same category as the tableau self:

sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Tableau tuples
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Standard tableau tuples
sage: StandardTableauTuples([[1],[2,2]])([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Standard tableau tuples
sage: StandardTableauTuples(level=2)([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Standard tableau tuples of level 2

to_chain()

Returns the chain of partitions corresponding to the standard tableau tuple self.

EXAMPLES:

sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).to_chain()
[([], []),
  ([], [1]),
  ([], [2]),
  ([], [2, 1]),
  ([], [2, 2]),
  ([1], [2, 2])]

class sage.combinat.tableau_tuple.StandardTableauTuples

Bases: sage.combinat.tableau_tuple.RowStandardTableauTuples

A factory class for the various classes of tuples of standard tableau.

INPUT:

There are three optional arguments:

• level – the level() of the tuples of tableaux

• size – the size() of the tuples of tableaux

• shape – a list or a partition tuple specifying the shape() of the standard tableau tuples

It is not necessary to use the keywords. If they are not used then the first integer argument specifies the level() and the second the size() of the tableau tuples.

OUTPUT:

The appropriate subclass of StandardTableauTuples.

A tuple of standard tableau is a tableau whose entries are positive integers which increase from left to right along the rows, and from top to bottom down the columns, in each component. The entries do NOT need to increase from left to right along the components.

Note

Sage uses the English convention for (tuples of) partitions and tableaux: the longer rows are displayed on top. As with PartitionTuple, in sage the cells, or nodes, of partition tuples are 0-based. For example, the (lexicographically) first cell in any non-empty partition tuple is \([0,0,0]\).

EXAMPLES:

sage: tabs=StandardTableauTuples([[3],[2,2]]); tabs
Standard tableau tuples of shape ([3], [2, 2])
sage: tabs.cardinality()
70
sage: tabs[10:16]
[([[1, 2, 3]], [[4, 6], [5, 7]]),
 ([[1, 2, 4]], [[3, 6], [5, 7]]),
 ([[1, 3, 4]], [[2, 6], [5, 7]]),
 ([[2, 3, 4]], [[1, 6], [5, 7]]),
 ([[1, 2, 5]], [[3, 6], [4, 7]]),
 ([[1, 3, 5]], [[2, 6], [4, 7]])]

sage: tabs=StandardTableauTuples(level=3); tabs
Standard tableau tuples of level 3
sage: tabs[100]
([[1, 2], [3]], [], [[4]])

sage: StandardTableauTuples()[0]
()

See also

• TableauTuples

• Tableau

• StandardTableau

• StandardTableauTuples

Element

alias of StandardTableauTuple

level_one_parent_class

alias of sage.combinat.tableau.StandardTableaux_all

shape()

Return the shape of the set of StandardTableauTuples, or None if it is not defined.

EXAMPLES:

sage: tabs=StandardTableauTuples(shape=[[5,2],[3,2],[],[1,1,1],[3]]); tabs
Standard tableau tuples of shape ([5, 2], [3, 2], [], [1, 1, 1], [3])
sage: tabs.shape()
([5, 2], [3, 2], [], [1, 1, 1], [3])
sage: StandardTableauTuples().shape() is None
True

class sage.combinat.tableau_tuple.StandardTableauTuples_all

Bases: sage.combinat.tableau_tuple.StandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Default class of all StandardTableauTuples with an arbitrary level() and size().

class sage.combinat.tableau_tuple.StandardTableauTuples_level(level)

Bases: sage.combinat.tableau_tuple.StandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all StandardTableauTuples with a fixed level and arbitrary size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: StandardTableauTuples(size=2).an_element()
([[1]], [[2]], [], [])
sage: StandardTableauTuples(size=4).an_element()
([[1]], [[2, 3, 4]], [], [])

class sage.combinat.tableau_tuple.StandardTableauTuples_level_size(level, size)

Bases: sage.combinat.tableau_tuple.StandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all StandardTableauTuples with a fixed level and a fixed size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: StandardTableauTuples(5,size=2).an_element()
([], [], [], [], [[1], [2]])
sage: StandardTableauTuples(2,size=4).an_element()
([[1]], [[2, 3], [4]])

cardinality()

Returns the number of elements in this set of tableaux.

EXAMPLES:

sage: StandardTableauTuples(3,2).cardinality()
12
sage: StandardTableauTuples(4,6).cardinality()
31936

class sage.combinat.tableau_tuple.StandardTableauTuples_shape(shape)

Bases: sage.combinat.tableau_tuple.StandardTableauTuples

Class of all StandardTableauTuples of a fixed shape.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: StandardTableauTuples([[2],[2,1]]).an_element()
([[2, 4]], [[1, 3], [5]])
sage: StandardTableauTuples([[10],[],[]]).an_element()
([[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], [], [])

cardinality()

Returns the number of standard Young tableau tuples of with the same shape as the partition tuple self.

Let \(\mu=(\mu^{(1)},\dots,\mu^{(l)})\) be the shape of the tableaux in self and let \(m_k=|\mu^{(k)}|\), for \(1\le k\le l\). Multiplying by a (unique) coset representative of the Young subgroup \(S_{m_1}\times\dots\times S_{m_l}\) inside the symmetric group \(S_n\), we can assume that \(t\) is standard and the numbers \(1,2...,n\) are entered in order from to right along the components of the tableau. Therefore, there are

\[\binom{n}{m_1,\dots,m_l}\prod_{k=1}^l |\text{Std}(\mu^{(k)})|\]

standard tableau tuples of this shape, where \(|\text{Std}(\mu^{(k)})|\) is the number of standard tableau of shape \(\mu^{(k)}\), for \(1 \leq k \leq l\). This is given by the hook length formula.

EXAMPLES:

sage: StandardTableauTuples([[3,2,1],[]]).cardinality()
16
sage: StandardTableauTuples([[1],[1],[1]]).cardinality()
6
sage: StandardTableauTuples([[2,1],[1],[1]]).cardinality()
40
sage: StandardTableauTuples([[3,2,1],[3,2]]).cardinality()
36960

last()

Returns the last standard tableau tuple in self, with respect to the order that they are generated by the iterator. This is just the standard tableau tuple with the numbers \(1,2, \ldots, n\), where \(n\) is size(), entered in order down the columns form right to left along the components.

EXAMPLES:

sage: StandardTableauTuples([[2],[2,2]]).last().pp()
  5  6     1  3
           2  4

random_element()

Returns a random standard tableau in self.

We do this by randomly selecting addable nodes to place \(1, 2, \ldots, n\). Of course we could do this recursively, but it’s more efficient to keep track of the (changing) list of addable nodes as we go.

EXAMPLES:

sage: StandardTableauTuples([[2],[2,1]]).random_element()  # random
([[1, 2]], [[3, 4], [5]])

class sage.combinat.tableau_tuple.StandardTableauTuples_size(size)

Bases: sage.combinat.tableau_tuple.StandardTableauTuples, sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

Class of all StandardTableauTuples with an arbitrary level and a fixed size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: StandardTableauTuples(size=2).an_element()
([[1]], [[2]], [], [])
sage: StandardTableauTuples(size=4).an_element()
([[1]], [[2, 3, 4]], [], [])

class sage.combinat.tableau_tuple.StandardTableaux_residue(residue)

Bases: sage.combinat.tableau_tuple.StandardTableauTuples

Class of all standard tableau tuples with a fixed residue sequence.

Implicitly, this also specifies the quantum characteristic, multicharge and hence the level and size of the tableaux.

Note

This class is not intended to be called directly, but rather, it is accessed through the standard tableaux.

EXAMPLES:

sage: StandardTableau([[1,2,3],[4,5]]).residue_sequence(2).standard_tableaux()
Standard tableaux with 2-residue sequence (0,1,0,1,0) and multicharge (0)
sage: StandardTableau([[1,2,3],[4,5]]).residue_sequence(3).standard_tableaux()
Standard tableaux with 3-residue sequence (0,1,2,2,0) and multicharge (0)
sage: StandardTableauTuple([[[5,6],[7]],[[1,2,3],[4]]]).residue_sequence(2,(0,0)).standard_tableaux()
Standard tableaux with 2-residue sequence (0,1,0,1,0,1,1) and multicharge (0,0)
sage: StandardTableauTuple([[[5,6],[7]],[[1,2,3],[4]]]).residue_sequence(3,(0,1)).standard_tableaux()
Standard tableaux with 3-residue sequence (1,2,0,0,0,1,2) and multicharge (0,1)

class sage.combinat.tableau_tuple.StandardTableaux_residue_shape(residue, shape)

Bases: sage.combinat.tableau_tuple.StandardTableaux_residue

All standard tableau tuples with a fixed residue and shape.

INPUT:

• shape – the shape of the partitions or partition tuples

• residue – the residue sequence of the label

EXAMPLES:

sage: res = StandardTableauTuple([[[1,3],[6]],[[2,7],[4],[5]]]).residue_sequence(3,(0,0))
sage: tabs = res.standard_tableaux([[2,1],[2,1,1]]); tabs
Standard (2,1|2,1^2)-tableaux with 3-residue sequence (0,0,1,2,1,2,1) and multicharge (0,0)
sage: tabs.shape()
([2, 1], [2, 1, 1])
sage: tabs.level()
2
sage: tabs[:6]
[([[2, 7], [6]], [[1, 3], [4], [5]]),
 ([[1, 7], [6]], [[2, 3], [4], [5]]),
 ([[2, 3], [6]], [[1, 7], [4], [5]]),
 ([[1, 3], [6]], [[2, 7], [4], [5]]),
 ([[2, 5], [6]], [[1, 3], [4], [7]]),
 ([[1, 5], [6]], [[2, 3], [4], [7]])]

an_element()

Return a particular element of self.

EXAMPLES:

sage: T = StandardTableau([[1,3],[2]]).residue_sequence(3).standard_tableaux([2,1])
sage: T.an_element()
[[1, 3], [2]]

class sage.combinat.tableau_tuple.TableauTuple(parent, t, check=True)

Bases: sage.combinat.combinat.CombinatorialElement

A class to model a tuple of tableaux.

INPUT:

• t – a list or tuple of Tableau, a list or tuple of lists of lists

OUTPUT:

• The Tableau tuple object constructed from t.

A TableauTuple is a tuple of tableau of shape a PartitionTuple. These combinatorial objects are useful is several areas of algebraic combinatorics. In particular, they are important in:

• the representation theory of the complex reflection groups of type \(G(l,1,n)\) and the representation theory of the associated (degenerate and non-degenerate) Hecke algebras. See, for example, [DJM1998]

• the crystal theory of (quantum) affine special linear groups and its integral highest weight modules and their canonical bases. See, for example, [BK2009].

These apparently different and unrelated contexts are, in fact, intimately related as in characteristic zero the cyclotomic Hecke algebras categorify the canonical bases of the integral highest weight modules of the quantum affine special linear groups.

The level() of a tableau tuple is the length of the tuples. This corresponds to the level of the corresponding highest weight module.

In sage a TableauTuple looks an behaves like a real tuple of (level 1) Tableaux. Many of the operations which are defined on Tableau extend to TableauTuples. Tableau tuples of level 1 are just ordinary Tableau.

In sage, the entries of Tableaux can be very general, including arbitrarily nested lists, so some lists can be interpreted either as a tuple of tableaux or simply as tableaux. If it is possible to interpret the input to TableauTuple as a tuple of tableaux then TableauTuple returns the corresponding tuple. Given a 1-tuple of tableaux the tableau itself is returned.

EXAMPLES:

sage: t = TableauTuple([ [[6,9,10],[11]], [[1,2,3],[4,5]], [[7],[8]] ]); t
([[6, 9, 10], [11]], [[1, 2, 3], [4, 5]], [[7], [8]])
sage: t.level()
3
sage: t.size()
11
sage: t.shape()
([3, 1], [3, 2], [1, 1])
sage: t.is_standard()
True
sage: t.pp() # pretty printing
 6  9 10     1  2  3     7
11           4  5        8
sage: t.category()
Category of elements of Tableau tuples
sage: t.parent()
Tableau tuples

sage: s = TableauTuple([ [['a','c','b'],['d','e']],[[(2,1)]]]); s
([['a', 'c', 'b'], ['d', 'e']], [[(2, 1)]])
sage: s.shape()
([3, 2], [1])
sage: s.size()
6

sage: TableauTuple([[],[],[]])  # The empty 3-tuple of tableaux
([], [], [])

sage: TableauTuple([[1,2,3],[4,5]])
[[1, 2, 3], [4, 5]]
sage: TableauTuple([[1,2,3],[4,5]]) == Tableau([[1,2,3],[4,5]])
True

See also

• StandardTableauTuple

• StandardTableauTuples

• StandardTableau

• StandardTableaux

• TableauTuple

• TableauTuples

• Tableau

• Tableaux

Element

alias of sage.combinat.tableau.Tableau

add_entry(cell, m)

Set the entry in cell equal to m. If the cell does not exist then extend the tableau, otherwise just replace the entry.

EXAMPLES:

sage: s = StandardTableauTuple([ [[3,4,7],[6,8]], [[9,13],[12]], [[1,5],[2,11],[10]] ]); s.pp()
  3  4  7     9 13     1  5
  6  8       12        2 11
                      10
sage: t = s.add_entry( (0,0,3),14); t.pp(); t.category()
  3  4  7 14     9 13     1  5
  6  8          12        2 11
                         10
Category of elements of Standard tableau tuples
sage: t = s.add_entry( (0,0,3),15); t.pp(); t.category()
  3  4  7 15     9 13     1  5
  6  8          12        2 11
                         10
Category of elements of Tableau tuples
sage: t = s.add_entry( (1,1,1),14); t.pp(); t.category()
  3  4  7     9 13     1  5
  6  8       12 14     2 11
                      10
Category of elements of Standard tableau tuples
sage: t = s.add_entry( (2,1,1),14); t.pp(); t.category()
  3  4  7     9 13     1  5
  6  8       12        2 14
                      10
Category of elements of Tableau tuples
sage: t = s.add_entry( (2,1,2),14); t.pp(); t.category()
Traceback (most recent call last):
...
IndexError: (2, 1, 2) is not an addable cell of the tableau

cells_containing(m)

Return the list of cells in which the letter m appears in the tableau self.

The list is ordered with cells appearing from left to right.

EXAMPLES:

sage: t = TableauTuple([[[4,5]],[[1,1,2,4],[2,4,4],[4]],[[1,3,4],[3,4]]])
sage: t.cells_containing(4)
[(0, 0, 0),
  (1, 2, 0),
  (1, 1, 1),
  (1, 1, 2),
  (1, 0, 3),
  (2, 1, 1),
  (2, 0, 2)]
sage: t.cells_containing(6)
[]

charge()

Return the charge of the reading word of self.

See charge() for more information.

EXAMPLES:

sage: TableauTuple([[[4,5]],[[1,1,2,4],[2,4,4],[4]],[[1,3,4],[3,4]]]).charge()
4

cocharge()

Return the cocharge of the reading word of self.

See cocharge() for more information.

EXAMPLES:

sage: TableauTuple([[[4,5]],[[1,1,2,4],[2,4,4],[4]],[[1,3,4],[3,4]]]).charge()
4

column_stabilizer()

Return the PermutationGroup corresponding to self. That is, return subgroup of the symmetric group of degree size() which is the column stabilizer of self.

EXAMPLES:

sage: cs = TableauTuple([[[1,2,3],[4,5]],[[6,7]],[[8],[9]]]).column_stabilizer()
sage: cs.order()
8
sage: PermutationGroupElement([(1,3,2),(4,5)]) in cs
False
sage: PermutationGroupElement([(1,4)]) in cs
True

components()

Return a list of the components of tableau tuple self. The \(components\) are the individual Tableau which are contained in the tuple self.

For compatibility with TableauTuples of level() 1, components() should be used to iterate over the components of TableauTuples.

EXAMPLES:

sage: for t in TableauTuple([[1,2,3],[4,5]]).components(): t.pp()
  1  2  3
  4  5
sage: for t in TableauTuple([ [[1,2,3],[4,5]], [[6,7],[8,9]] ]).components(): t.pp()
  1  2  3
  4  5
  6  7
  8  9

conjugate()

Return the conjugate of the tableau tuple self.

The conjugate tableau tuple \(T'\) is the TableauTuple obtained from \(T\) by reversing the order of the components and conjugating each component – that is, swapping the rows and columns of the all of Tableau in \(T\) (see sage.combinat.tableau.Tableau.conjugate()).

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).conjugate()
([[9, 11, 12], [10]], [[5, 8], [6], [7]], [[1, 3], [2, 4]])

content(k, multicharge)

Return the content k in self.

The content of \(k\) in a standard tableau. That is, if \(k\) appears in row \(r\) and column \(c\) of the tableau, then we return \(c - r + a_k\), where the multicharge is \((a_1, a_2, \ldots, a_l)\) and \(l\) is the level of the tableau.

The multicharge determines the dominant weight

\[\Lambda = \sum_{i=1}^l \Lambda_{a_i}\]

of the affine special linear group. In the combinatorics, the multicharge simply offsets the contents in each component so that the cell \((k, r, c)\) has content \(a_k + c - r\).

INPUT:

• k – an integer in \(\{1, 2, \ldots, n\}\)

• multicharge – a sequence of integers of length \(l\)

Here \(l\) is the level() and \(n\) is the size() of self.

EXAMPLES:

sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).content(3,[0,0])
-1
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).content(3,[0,1])
0
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).content(3,[0,2])
1
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).content(6,[0,2])
Traceback (most recent call last):
...
ValueError: 6 must be contained in the tableaux

entries()

Return a sorted list of all entries of self, in the order obtained by reading across the rows.

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).entries()
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
sage: TableauTuple([[[1,2],[3,4]],[[9,10],[11],[12]],[[5,6,7],[8]]]).entries()
[1, 2, 3, 4, 9, 10, 11, 12, 5, 6, 7, 8]

entry(l, r, c)

Return the entry of the cell (l, r, c) in self.

A cell is a tuple (l, r, c) of coordinates, where l is the component index, r is the row index, and c is the column index.

EXAMPLES:

sage: t = TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]])
sage: t.entry(1, 0, 0)
5
sage: t.entry(1, 1, 1)
Traceback (most recent call last):
...
IndexError: tuple index out of range

first_column_descent()

Return the first cell of self is not column standard.

Cells are ordered left to right along the rows and then top to bottom. That is, return the cell \((k,r,c)\) with \((k,r,c)\) minimal such that the entry in position \((k,r,c)\) is bigger than the entry in position \((k,r,c+1)\). If there is no such cell then None is returned - in this case the tableau is column strict.

OUTPUT:

The cell corresponding to the first column descent or None if the tableau is column strict.

EXAMPLES:

sage: TableauTuple([[[3,5,6],[2,4,5]],[[1,4,5],[2,3]]]).first_column_descent()
(0, 0, 0)
sage: Tableau([[[1,2,3],[4]],[[5,6,7],[8,9]]]).first_column_descent() is None
True

first_row_descent()

Return the first cell of self that is not row standard.

Cells are ordered left to right along the rows and then top to bottom. That is, the cell minimal \((k,r,c)\) such that the entry in position \((k,r,c)\) is bigger than the entry in position \((k,r,c+1)\). If there is no such cell then None is returned - in this case the tableau is row strict.

OUTPUT:

The cell corresponding to the first row descent or None if the tableau is row strict.

EXAMPLES:

sage: TableauTuple([[[5,6,7],[1,2]],[[1,3,2],[4]]]).first_row_descent()
(1, 0, 1)
sage: TableauTuple([[[1,2,3],[4]],[[6,7,8],[1,2,3]],[[1,11]]]).first_row_descent() is None
True

is_column_strict()

Return True if the tableau self is column strict and False otherwise.

A tableau tuple is column strict if the entries in each column of each component are in increasing order, when read from top to bottom.

EXAMPLES:

sage: TableauTuple([[[5,7],[8]],[[1, 3], [2, 4]],[[6]]]).is_column_strict()
True
sage: TableauTuple([[[1, 2], [2, 4]],[[4,5,6],[7,8]]]).is_column_strict()
True
sage: TableauTuple([[[1]],[[2, 3], [2, 4]]]).is_column_strict()
False
sage: TableauTuple([[[1]],[[2, 2], [4,5]]]).is_column_strict()
True
sage: TableauTuple([[[1,2],[6,7]],[[4,8], [6, 9]],[]]).is_column_strict()
True

is_row_strict()

Return True if the tableau self is row strict and False otherwise.

A tableau tuple is row strict if the entries in each row of each component are in increasing order, when read from left to right.

EXAMPLES:

sage: TableauTuple([[[5,7],[8]],[[1, 3], [2, 4]],[[6]]]).is_row_strict()
True
sage: TableauTuple([[[1, 2], [2, 4]],[[4,5,6],[7,8]]]).is_row_strict()
True
sage: TableauTuple([[[1]],[[2, 3], [2, 4]]]).is_row_strict()
True
sage: TableauTuple([[[1]],[[2, 2], [4,5]]]).is_row_strict()
False
sage: TableauTuple([[[1,2],[6,7]],[[4,8], [6, 9]],[]]).is_row_strict()
True

is_standard()

Return True if the tableau self is a standard tableau and False otherwise.

A tableau tuple is standard if it is row standard, column standard and the entries in the tableaux are \(1, 2, \ldots, n\), where \(n\) is the size() of the underlying partition tuple of self.

EXAMPLES:

sage: TableauTuple([[[5,7],[8]],[[1, 3], [2, 4]],[[6]]]).is_standard()
True
sage: TableauTuple([[[1, 2], [2, 4]],[[4,5,6],[7,8]]]).is_standard()
False
sage: TableauTuple([[[1]],[[2, 3], [2, 4]]]).is_standard()
False
sage: TableauTuple([[[1]],[[2, 2], [4,5]]]).is_row_strict()
False
sage: TableauTuple([[[1,2],[6,7]],[[4,8], [6, 9]],[]]).is_standard()
False

level()

Return the level of the tableau self, which is just the number of components in the tableau tuple self.

EXAMPLES:

sage: TableauTuple([[[7,8,9]],[],[[1,2,3],[4,5],[6]]]).level()
3

pp()

Pretty printing for the tableau tuple self.

EXAMPLES:

sage: TableauTuple([ [[1,2,3],[4,5]], [[1,2,3],[4,5]] ]).pp()
  1  2  3     1  2  3
  4  5        4  5
sage: TableauTuple([ [[1,2],[3],[4]],[],[[6,7,8],[10,11],[12],[13]]]).pp()
  1  2     -     6  7  8
  3             10 11
  4             12
                13
sage: t = TableauTuple([ [[1,2,3],[4,5],[6],[9]], [[1,2,3],[4,5,8]], [[11,12,13],[14]] ])
sage: t.pp()
  1  2  3     1  2  3    11 12 13
  4  5        4  5  8    14
  6
  9
sage: TableauTuples.options(convention="french")
sage: t.pp()
  9
  6
  4  5        4  5  8    14
  1  2  3     1  2  3    11 12 13
sage: TableauTuples.options._reset()

reduced_column_word()

Return the lexicographically minimal reduced expression for the permutation that maps the initial_column_tableau() to self.

This reduced expression is a minimal length coset representative for the corresponding Young subgroup. In one line notation, the permutation is obtained by concatenating the rows of the tableau from top to bottom in each component, and then left to right along the components.

EXAMPLES:

sage: StandardTableauTuple([[[7,9],[8]],[[1,4,6],[2,5],[3]]]).reduced_column_word()
[]
sage: StandardTableauTuple([[[7,9],[8]],[[1,3,6],[2,5],[4]]]).reduced_column_word()
[3]
sage: StandardTableauTuple([[[6,9],[8]],[[1,3,7],[2,5],[4]]]).reduced_column_word()
[3, 6]
sage: StandardTableauTuple([[[6,8],[9]],[[1,3,7],[2,5],[4]]]).reduced_column_word()
[3, 6, 8]
sage: StandardTableauTuple([[[5,8],[9]],[[1,3,7],[2,6],[4]]]).reduced_column_word()
[3, 6, 5, 8]

reduced_row_word()

Return the lexicographically minimal reduced expression for the permutation that maps the initial_tableau() to self.

This reduced expression is a minimal length coset representative for the corresponding Young subgroup. In one line notation, the permutation is obtained by concatenating the rows of the tableau from top to bottom in each component, and then left to right along the components.

EXAMPLES:

sage: StandardTableauTuple([[[1,2],[3]],[[4,5,6],[7,8],[9]]]).reduced_row_word()
[]
sage: StandardTableauTuple([[[1,2],[3]],[[4,5,6],[7,9],[8]]]).reduced_row_word()
[8]
sage: StandardTableauTuple([[[1,2],[3]],[[4,5,7],[6,9],[8]]]).reduced_row_word()
[6, 8]
sage: StandardTableauTuple([[[1,2],[3]],[[4,5,8],[6,9],[7]]]).reduced_row_word()
[6, 8, 7]
sage: StandardTableauTuple([[[1,2],[3]],[[4,5,9],[6,8],[7]]]).reduced_row_word()
[6, 7, 8, 7]
sage: StandardTableauTuple([[[7,9],[8]],[[1,3,5],[2,6],[4]]]).reduced_row_word()
[2, 3, 2, 1, 4, 3, 2, 5, 4, 3, 6, 5, 4, 3, 2, 7, 6, 5, 8, 7, 6, 5, 4]

residue(k, e, multicharge)

Return the residue of the integer k in the tableau self.

The residue of \(k\) is \(c - r + a_k\) in \(\ZZ / e\ZZ\), where \(k\) appears in row \(r\) and column \(c\) of the tableau and the multicharge is \((a_1, a_2, \ldots, a_l)\).

The multicharge determines the dominant weight

\[\sum_{i=1}^l \Lambda_{a_i}\]

for the affine special linear group. In the combinatorics, it simply offsets the contents in each component so that the cell \((k, 0, 0)\) has content \(a_k\).

INPUT:

• k – an integer in \(\{1, 2, \ldots, n\}\)

• e – an integer in \(\{0, 2, 3, 4, 5, \ldots\}\)

• multicharge – a list of integers of length \(l\)

Here \(l\) is the level() and \(n\) is the size() of self.

OUTPUT:

The residue of k in a standard tableau. That is,

EXAMPLES:

sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue(1, 3,[0,0])
0
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue(1, 3,[0,1])
1
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue(1, 3,[0,2])
2
sage: StandardTableauTuple([[[5]],[[1,2],[3,4]]]).residue(6, 3,[0,2])
Traceback (most recent call last):
...
ValueError: 6 must be contained in the tableaux

restrict(m=None)

Returns the restriction of the standard tableau self to m.

The restriction is the subtableau of self whose entries are less than or equal to m.

By default, m is one less than the current size.

EXAMPLES:

sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict()
([], [[1, 2], [3, 4]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(6)
([[5]], [[1, 2], [3, 4]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(5)
([[5]], [[1, 2], [3, 4]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(4)
([], [[1, 2], [3, 4]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3)
([], [[1, 2], [3]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(2)
([], [[1, 2]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(1)
([], [[1]])
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(0)
([], [])

Where possible the restricted tableau belongs to the same category as the original tableaux:

sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Tableau tuples
sage: TableauTuple([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Tableau tuples
sage: TableauTuples(level=2)([[[5]],[[1,2],[3,4]]]).restrict(3).category()
Category of elements of Tableau tuples of level 2

row_stabilizer()

Return the PermutationGroup corresponding to self. That is, return subgroup of the symmetric group of degree size() which is the row stabilizer of self.

EXAMPLES:

sage: rs = TableauTuple([[[1,2,3],[4,5]],[[6,7]],[[8],[9]]]).row_stabilizer()
sage: rs.order()
24
sage: PermutationGroupElement([(1,3,2),(4,5)]) in rs
True
sage: PermutationGroupElement([(1,4)]) in rs
False
sage: rs.one().domain()
[1, 2, 3, 4, 5, 6, 7, 8, 9]

shape()

Returns the PartitionTuple which is the shape of the tableau tuple self.

EXAMPLES:

sage: TableauTuple([[[7,8,9]],[],[[1,2,3],[4,5],[6]]]).shape()
([3], [], [3, 2, 1])

size()

Returns the size of the tableau tuple self, which is just the number of boxes, or the size, of the underlying PartitionTuple.

EXAMPLES:

sage: TableauTuple([[[7,8,9]],[],[[1,2,3],[4,5],[6]]]).size()
9

symmetric_group_action_on_entries(w)

Return the action of a permutation w on self.

Consider a standard tableau tuple \(T = (t^{(1)}, t^{(2)}, \ldots t^{(l)})\) of size \(n\), then the action of \(w \in S_n\) is defined by permuting the entries of \(T\) (recall they are \(1, 2, \ldots, n\)). In particular, suppose the entry at cell \((k, i, j)\) is \(a\), then the entry becomes \(w(a)\). In general, the resulting tableau tuple \(wT\) may not be standard.

INPUT:

• w – a permutation

EXAMPLES:

sage: TableauTuple([[[1,2],[4]],[[3,5]]]).symmetric_group_action_on_entries( Permutation(((4,5))) )
([[1, 2], [5]], [[3, 4]])
sage: TableauTuple([[[1,2],[4]],[[3,5]]]).symmetric_group_action_on_entries( Permutation(((1,2))) )
([[2, 1], [4]], [[3, 5]])

to_list()

Return the list representation of the tableaux tuple self.

EXAMPLES:

sage: TableauTuple([ [[1,2,3],[4,5]], [[6,7],[8,9]] ]).to_list()
[[[1, 2, 3], [4, 5]], [[6, 7], [8, 9]]]

to_permutation()

Returns a permutation with the entries in the tableau tuple self which is obtained by self obtained by reading the entries of the tableau tuple in order from left to right along the rows, and then top to bottom, in each component and then left to right along the components.

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).to_permutation()
[12, 11, 9, 10, 8, 5, 6, 7, 3, 4, 1, 2]

to_word()

Returns a word obtained from a row reading of the tableau tuple self.

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).to_word_by_row()
word: 12,11,9,10,8,5,6,7,3,4,1,2

to_word_by_column()

Returns the word obtained from a column reading of the tableau tuple self.

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).to_word_by_column()
word: 12,11,9,10,8,5,6,7,3,1,4,2

to_word_by_row()

Returns a word obtained from a row reading of the tableau tuple self.

EXAMPLES:

sage: TableauTuple([[[1,2],[3,4]],[[5,6,7],[8]],[[9,10],[11],[12]]]).to_word_by_row()
word: 12,11,9,10,8,5,6,7,3,4,1,2

up(n=None)

An iterator for all the TableauTuple that can be obtained from self by adding a cell with the label n. If n is not specified then a cell with label n will be added to the tableau tuple, where n-1 is the size of the tableau tuple before any cells are added.

EXAMPLES:

sage: list(TableauTuple([[[1,2]],[[3]]]).up())
[([[1, 2, 4]], [[3]]),
  ([[1, 2], [4]], [[3]]),
  ([[1, 2]], [[3, 4]]),
  ([[1, 2]], [[3], [4]])]

class sage.combinat.tableau_tuple.TableauTuples

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

A factory class for the various classes of tableau tuples.

INPUT:

There are three optional arguments:

• shape – determines a PartitionTuple which gives the shape of the TableauTuples

• level – the level of the tableau tuples (positive integer)

• size – the size of the tableau tuples (non-negative integer)

It is not necessary to use the keywords. If they are not specified then the first integer argument specifies the level and the second the size of the tableaux.

OUTPUT:

• The corresponding class of tableau tuples.

The entries of a tableau can be any sage object. Because of this, no enumeration of the set of TableauTuples is possible.

EXAMPLES:

sage: T3 = TableauTuples(3); T3
Tableau tuples of level 3
sage: [['a','b']] in TableauTuples()
True
sage: [['a','b']] in TableauTuples(level=3)
False
sage: t = TableauTuples(level=3)([[],[[1,1,1]],[]]); t
([], [[1, 1, 1]], [])
sage: t in T3
True
sage: t in TableauTuples()
True
sage: t in TableauTuples(size=3)
True
sage: t in TableauTuples(size=4)
False
sage: t in StandardTableauTuples()
False
sage: t.parent()
Tableau tuples of level 3
sage: t.category()
Category of elements of Tableau tuples of level 3

See also

• Tableau

• StandardTableau

• StandardTableauTuples

Element

alias of TableauTuple

level()

Return the level of a tableau tuple in self, or None if different tableau tuples in self can have different sizes. The level of a tableau tuple is just the level of the underlying PartitionTuple.

EXAMPLES:

sage: TableauTuples().level() is None
True
sage: TableauTuples(7).level()
7

level_one_parent_class

alias of sage.combinat.tableau.Tableaux_all

list()

If the set of tableau tuples self is finite then this function returns the list of these tableau tuples. If the class is infinite an error is returned.

EXAMPLES:

sage: StandardTableauTuples([[2,1],[2]]).list()
[([[1, 2], [3]], [[4, 5]]),
 ([[1, 3], [2]], [[4, 5]]),
 ([[1, 2], [4]], [[3, 5]]),
 ([[1, 3], [4]], [[2, 5]]),
 ([[2, 3], [4]], [[1, 5]]),
 ([[1, 4], [2]], [[3, 5]]),
 ([[1, 4], [3]], [[2, 5]]),
 ([[2, 4], [3]], [[1, 5]]),
 ([[1, 2], [5]], [[3, 4]]),
 ([[1, 3], [5]], [[2, 4]]),
 ([[2, 3], [5]], [[1, 4]]),
 ([[1, 4], [5]], [[2, 3]]),
 ([[2, 4], [5]], [[1, 3]]),
 ([[3, 4], [5]], [[1, 2]]),
 ([[1, 5], [2]], [[3, 4]]),
 ([[1, 5], [3]], [[2, 4]]),
 ([[2, 5], [3]], [[1, 4]]),
 ([[1, 5], [4]], [[2, 3]]),
 ([[2, 5], [4]], [[1, 3]]),
 ([[3, 5], [4]], [[1, 2]])]

options(*get_value, **set_value)

Sets the global options for elements of the tableau, skew_tableau, and tableau tuple classes. The defaults are for tableau to be displayed as a list, latexed as a Young diagram using the English convention.

OPTIONS:

• ascii_art – (default: repr) Controls the ascii art output for tableaux

  • compact – minimal length ascii art

  • repr – display using the diagram string representation

  • table – display as a table

• convention – (default: English) Sets the convention used for displaying tableaux and partitions

  • English – use the English convention

  • French – use the French convention

• display – (default: list) Controls the way in which tableaux are printed

  • array – alias for diagram

  • compact – minimal length string representation

  • diagram – display as Young diagram (similar to pp()

  • ferrers_diagram – alias for diagram

  • list – print tableaux as lists

  • young_diagram – alias for diagram

• latex – (default: diagram) Controls the way in which tableaux are latexed

  • array – alias for diagram

  • diagram – as a Young diagram

  • ferrers_diagram – alias for diagram

  • list – as a list

  • young_diagram – alias for diagram

• notation – alternative name for convention

Note

Changing the convention for tableaux also changes the convention for partitions.

If no parameters are set, then the function returns a copy of the options dictionary.

EXAMPLES:

sage: T = Tableau([[1,2,3],[4,5]])
sage: T
[[1, 2, 3], [4, 5]]
sage: Tableaux.options.display="array"
sage: T
  1  2  3
  4  5
sage: Tableaux.options.convention="french"
sage: T
  4  5
  1  2  3

Changing the convention for tableaux also changes the convention for partitions and vice versa:

sage: P = Partition([3,3,1])
sage: print(P.ferrers_diagram())
*
***
***
sage: Partitions.options.convention="english"
sage: print(P.ferrers_diagram())
***
***
*
sage: T
  1  2  3
  4  5

The ASCII art can also be changed:

sage: t = Tableau([[1,2,3],[4,5]])
sage: ascii_art(t)
  1  2  3
  4  5
sage: Tableaux.options.ascii_art = "table"
sage: ascii_art(t)
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 |
+---+---+
sage: Tableaux.options.ascii_art = "compact"
sage: ascii_art(t)
|1|2|3|
|4|5|
sage: Tableaux.options._reset()

See GlobalOptions for more features of these options.

size()

Return the size of a tableau tuple in self, or None if different tableau tuples in self can have different sizes. The size of a tableau tuple is just the size of the underlying PartitionTuple.

EXAMPLES:

sage: TableauTuples(size=14).size()
14

class sage.combinat.tableau_tuple.TableauTuples_all

Bases: sage.combinat.tableau_tuple.TableauTuples

The parent class of all TableauTuples, with arbitrary level and size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: TableauTuples().an_element()
([[1]], [[2]], [[3]], [[4]], [[5]], [[6]], [[7]])

class sage.combinat.tableau_tuple.TableauTuples_level(level)

Bases: sage.combinat.tableau_tuple.TableauTuples

Class of all TableauTuples with a fixed level and arbitrary size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: TableauTuples(3).an_element()
([], [], [])
sage: TableauTuples(5).an_element()
([], [], [], [], [])
sage: T = TableauTuples(0)
Traceback (most recent call last):
...
ValueError: the level must be a positive integer

class sage.combinat.tableau_tuple.TableauTuples_level_size(level, size)

Bases: sage.combinat.tableau_tuple.TableauTuples

Class of all TableauTuples with a fixed level and a fixed size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: TableauTuples(3,0).an_element()
([], [], [])
sage: TableauTuples(3,1).an_element()
([[1]], [], [])
sage: TableauTuples(3,2).an_element()
([[1, 2]], [], [])

class sage.combinat.tableau_tuple.TableauTuples_size(size)

Bases: sage.combinat.tableau_tuple.TableauTuples

Class of all TableauTuples with a arbitrary level and fixed size.

an_element()

Returns a particular element of the class.

EXAMPLES:

sage: TableauTuples(size=3).an_element()
([], [[1, 2, 3]], [])
sage: TableauTuples(size=0).an_element()
([], [], [])