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:
Factory classes:
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.
-
sage.combinat.tableau_tuple.
RowStandardTableauTuple
¶ 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 aPartitionTuple
.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 aRowStandardTableauTuple
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 fromt
.
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 aTableauTuple
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
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples
¶ A factory class for the various classes of tuples of row standard tableau.
INPUT:
There are three optional arguments:
level
– thelevel()
of the tuples of tableauxsize
– thesize()
of the tuples of tableauxshape
– a list or a partition tuple specifying theshape()
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 thesize()
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] ([])
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_all
¶ Default class of all
RowStandardTableauTuples
with an arbitrarylevel()
andsize()
.
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_level
¶ Class of all
RowStandardTableauTuples
with a fixedlevel
and arbitrarysize
.
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_level_size
¶ Class of all
RowStandardTableauTuples
with a fixedlevel
and a fixedsize
.
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_residue
¶ 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)
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_residue_shape
¶ All row standard tableau tuples with a fixed residue and shape.
INPUT:
shape
– the shape of the partitions or partition tuplesresidue
– 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]])]
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_shape
¶ Class of all
RowStandardTableauTuples
of a fixed shape.
-
sage.combinat.tableau_tuple.
RowStandardTableauTuples_size
¶ Class of all
RowStandardTableauTuples
with an arbitrarylevel
and a fixedsize
.
-
sage.combinat.tableau_tuple.
StandardTableauTuple
¶ 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() TrueThe 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 aStandardTableauTuple
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 fromt
.
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 aTableauTuple
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]])
-
sage.combinat.tableau_tuple.
StandardTableauTuples
¶ A factory class for the various classes of tuples of standard tableau.
INPUT:
There are three optional arguments:
level
– thelevel()
of the tuples of tableauxsize
– thesize()
of the tuples of tableauxshape
– a list or a partition tuple specifying theshape()
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 thesize()
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] ()
-
sage.combinat.tableau_tuple.
StandardTableauTuples_all
¶ Default class of all
StandardTableauTuples
with an arbitrarylevel()
andsize()
.
-
sage.combinat.tableau_tuple.
StandardTableauTuples_level
¶ Class of all
StandardTableauTuples
with a fixedlevel
and arbitrarysize
.
-
sage.combinat.tableau_tuple.
StandardTableauTuples_level_size
¶ Class of all
StandardTableauTuples
with a fixedlevel
and a fixedsize
.
-
sage.combinat.tableau_tuple.
StandardTableauTuples_shape
¶ Class of all
StandardTableauTuples
of a fixed shape.
-
sage.combinat.tableau_tuple.
StandardTableauTuples_size
¶ Class of all
StandardTableauTuples
with an arbitrarylevel
and a fixedsize
.
-
sage.combinat.tableau_tuple.
StandardTableaux_residue
¶ 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)
-
sage.combinat.tableau_tuple.
StandardTableaux_residue_shape
¶ All standard tableau tuples with a fixed residue and shape.
INPUT:
shape
– the shape of the partitions or partition tuplesresidue
– 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]])]
-
sage.combinat.tableau_tuple.
TableauTuple
¶ A class to model a tuple of tableaux.
INPUT:
t
– a list or tuple ofTableau
, 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 aPartitionTuple
. 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 onTableau
extend toTableauTuples
. Tableau tuples of level 1 are just ordinaryTableau
.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 toTableauTuple
as a tuple of tableaux thenTableauTuple
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
-
sage.combinat.tableau_tuple.
TableauTuples
¶ A factory class for the various classes of tableau tuples.
INPUT:
There are three optional arguments:
shape
– determines aPartitionTuple
which gives the shape of theTableauTuples
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 thesize
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
-
sage.combinat.tableau_tuple.
TableauTuples_all
¶ The parent class of all
TableauTuples
, with arbitrarylevel
andsize
.
-
sage.combinat.tableau_tuple.
TableauTuples_level
¶ Class of all
TableauTuples
with a fixedlevel
and arbitrarysize
.
-
sage.combinat.tableau_tuple.
TableauTuples_level_size
¶ Class of all
TableauTuples
with a fixedlevel
and a fixedsize
.
-
sage.combinat.tableau_tuple.
TableauTuples_size
¶ Class of all
TableauTuples
with a arbitrarylevel
and fixedsize
.