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# 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 tulpes 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).

This file consists of the following major classes:

Element classes:

- `TableauTuples`
- `StandardTableauTuples`

Factory classes:

- `TableauTuples`
- `StandardTableauTuples`

Parent classes:

- `TableauTuples_all`
- `TableauTuples_level`
- `TableauTuples_size`
- `TableauTuples_level_size`
- `StandardTableauTuples_all`
- `StandardTableauTuples_level`
- `StandardTableauTuples_size`
- `StandardTableauTuples_level_size`
- `StandardTableauTuples_shape`

See also

- `Tableau`
- `StandardTableau`
- `Tableaux`
- `StandardTableaux`
- `Partitions`
- `PartitionTuples`

Todo

Implement semistandard tableau tuples as defined in [DJM].

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

REFERENCES:

[DJM] | (1, 2) R. Dipper, G. James and A. Mathas “The cyclotomic q-Schur algebra”, Math. Z, 229 (1999), 385-416.

[BK] | J. Brundan and A. Kleshchev “Graded decomposition numbers for cyclotomic Hecke algebras”, Adv. Math., 222 (2009), 1883-1942”

class `sage.combinat.tableau_tuple.``StandardTableauTuple`(parent, t)

  Bases: `sage.combinat.tableau_tuple.TableauTuple`

  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.

  Note

  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 coorindates `[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 print
      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`

  TESTS:

      sage: StandardTableauTuple( [[1,2,3],[4,5]] ).category()  # indirect doctest
      Category of elements of Standard tableaux
      sage: StandardTableauTuple([[[1,2,3],[4,5]]]).category()  # indirect doctest
      Category of elements of Standard tableaux
      sage: StandardTableauTuples()([[[1,2,3],[4,5]]]).category()  # indirect doctest
      Category of elements of Standard tableaux

      sage: StandardTableauTuple([[[1,2,3]],[[1]]])
      Traceback (most recent call last):
      ...
      ValueError: entries must be in bijection with {1,2,...,n}

      sage: StandardTableauTuple([[],[[1,2,1]]])
      Traceback (most recent call last):
      ...
      ValueError: tableaux must be row strict

      sage: StandardTableauTuple([ [[1,2,4],[6]],[[0,1]],[[10]] ])
      Traceback (most recent call last):
      ...
      ValueError: entries must be in bijection with {1,2,...,n}

      sage: TestSuite(  StandardTableauTuple([[[1,3,4],[6]],[[2],[5]]]) ).run()
      sage: TestSuite(  StandardTableauTuple([[[1,3,4],[6]],[], [[2],[5]]]) ).run()
      sage: TestSuite(  StandardTableauTuple([[[1,3,4],[6]],[[7]], [[2],[5]]]) ).run()

  `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 \(c-r\) + `multicharge[k]`.

    The `multicharge` = \([m_1, \ldots, m_l]\) determines the dominant weight

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

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

    INPUT:

    - `k` – An integer with \(1 \leq k \leq n\)
    - `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

  `dominates`(t)

    Returns `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

  `inverse`(k)

    Return the cell containing `k` in the tableau tuple `self`.

    EXAMPLES:

        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)

  `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.TableauTuples`

  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 then 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 down 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([[2],[1],[1,1]]); tabs
      Standard tableau tuples of shape ([2], [1], [1, 1])
      sage: tabs.cardinality()
      30
      sage: tabs.list()
      [([[1, 2]], [[3]], [[4], [5]]),
       ([[1, 3]], [[2]], [[4], [5]]),
       ([[2, 3]], [[1]], [[4], [5]]),
       ([[1, 2]], [[4]], [[3], [5]]),
       ([[1, 3]], [[4]], [[2], [5]]),
       ([[2, 3]], [[4]], [[1], [5]]),
       ([[1, 4]], [[2]], [[3], [5]]),
       ([[2, 4]], [[1]], [[3], [5]]),
       ([[1, 4]], [[3]], [[2], [5]]),
       ([[2, 4]], [[3]], [[1], [5]]),
       ([[3, 4]], [[1]], [[2], [5]]),
       ([[3, 4]], [[2]], [[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]]),
       ([[2, 5]], [[1]], [[3], [4]]),
       ([[1, 5]], [[3]], [[2], [4]]),
       ([[2, 5]], [[3]], [[1], [4]]),
       ([[3, 5]], [[1]], [[2], [4]]),
       ([[3, 5]], [[2]], [[1], [4]]),
       ([[1, 5]], [[4]], [[2], [3]]),
       ([[2, 5]], [[4]], [[1], [3]]),
       ([[3, 5]], [[4]], [[1], [2]]),
       ([[4, 5]], [[1]], [[2], [3]]),
       ([[4, 5]], [[2]], [[1], [3]]),
       ([[4, 5]], [[3]], [[1], [2]])]

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

      sage: StandardTableauTuples()[0]
      ()

  TESTS:

      sage: TestSuite( StandardTableauTuples() ).run()
      sage: TestSuite( StandardTableauTuples(level=1) ).run()
      sage: TestSuite( StandardTableauTuples(level=4) ).run()
      sage: TestSuite( StandardTableauTuples(size=0) ).run(max_runs=50) # recursion depth exceeded with default max_runs
      sage: TestSuite( StandardTableauTuples(size=6) ).run()
      sage: TestSuite( StandardTableauTuples(level=1, size=0) ).run()
      sage: TestSuite( StandardTableauTuples(level=1, size=0) ).run()
      sage: TestSuite( StandardTableauTuples(level=1, size=10) ).run()
      sage: TestSuite( StandardTableauTuples(level=4, size=0) ).run()
      sage: TestSuite( StandardTableauTuples(level=4, size=0) ).run()
      sage: TestSuite( StandardTableauTuples(level=4, size=10) ).run() # long time
      sage: TestSuite( StandardTableauTuples(shape=[[1],[3,1],[],[2,1]]) ).run()

  See also

  - `TableauTuples`
  - `Tableau`
  - `StandardTableau`
  - `StandardTableauTuples`
  `Element`

    alias of `StandardTableauTuple`

  `an_element`()

    Returns a particular element of the class.

    EXAMPLES:

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

  `level_one_parent_class`

    alias of `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`

  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`

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

  `an_element`()

    Returns a particular element of the class.

    EXAMPLES:

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

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

  Bases: `sage.combinat.tableau_tuple.StandardTableauTuples`

  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: t=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`

  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.``TableauTuple`(parent, t)

  Bases: `sage.combinat.combinat.CombinatorialObject`, `sage.structure.element.Element`

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

  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 tulpes 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-tulpe 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 print
      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`

  TESTS:

      sage: TableauTuple( [[1,2,3],[4,5]] ).category()
      Category of elements of Tableaux
      sage: TableauTuple([[[1,2,3],[4,5]]]).category()
      Category of elements of Tableaux

      sage: TableauTuple([[1],[2,3]])
      Traceback (most recent call last):
      ...
      ValueError: A tableau must be a list of lists.

      sage: TestSuite( TableauTuple([ [[1,2],[3,4]], [[1,2],[3,4]] ]) ).run()
      sage: TestSuite( TableauTuple([ [[1,2],[3,4]], [], [[1,2],[3,4]] ]) ).run()
      sage: TestSuite( TableauTuple([[[1,1],[1]],[[1,1,1]],[[1],[1],[1]],[[1]]]) ).run()

  `Element`

    alias of `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)

    Returns 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`()

    Returns the conjugate of the tableau tuple `self`. That is, the `TableauTuple` obtained from `self` by reversing the order of the components and conjugating each component – that is, swapping the rows and columns of the all of `Tableau` in `self` (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]])

  `entries`()

    Returns 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)

    Returns 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: list index out of range

  `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`()

    Returns `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.global_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.global_options.reset()

  `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

  TESTS:

      sage: TableauTuples(0)
      Traceback (most recent call last):
      ...
      ValueError: the level must be a positive integer

      sage: t = TableauTuples(3)([[],[],[[1,2],[3]]])
      sage: t.parent()
      Tableau tuples of level 3
      sage: TableauTuples(t)
      Traceback (most recent call last):
      ...
      ValueError: the level must be a positive integer
      sage: TableauTuples(3)([[1, 1]])
      Traceback (most recent call last):
      ...
      ValueError: [[1, 1]] is not an element of Tableau tuples of level 3

      sage: t0 = Tableau([[1]])
      sage: t1 = TableauTuples()([[1]])
      sage: t2 = TableauTuples()(t1)
      sage: t0 == t1 == t2
      True
      sage: t1 in TableauTuples()
      True
      sage: t1 in TableauTuples(1)
      True
      sage: t1 in TableauTuples(2)
      False

      sage: [[1]] in TableauTuples()
      True
      sage: [] in TableauTuples()
      True

      sage: TableauTuples(level=0)
      Traceback (most recent call last):
      ...
      ValueError: the level must be a positive integer

      sage: TestSuite( TableauTuples() ).run()
      sage: TestSuite( TableauTuples(level=1) ).run()
      sage: TestSuite( TableauTuples(level=2) ).run()
      sage: TestSuite( TableauTuples(level=6) ).run()
      sage: TestSuite( TableauTuples(size=0) ).run()
      sage: TestSuite( TableauTuples(size=1) ).run()
      sage: TestSuite( TableauTuples(size=2) ).run()
      sage: TestSuite( TableauTuples(size=10) ).run()
      sage: TestSuite( TableauTuples(level=1, size=0) ).run()
      sage: TestSuite( TableauTuples(level=1, size=1) ).run()
      sage: TestSuite( TableauTuples(level=1, size=10) ).run()
      sage: TestSuite( TableauTuples(level=2, size=0) ).run()
      sage: TestSuite( TableauTuples(level=2, size=1) ).run()
      sage: TestSuite( TableauTuples(level=2, size=10) ).run()
      sage: TestSuite( TableauTuples(level=6, size=0) ).run()
      sage: TestSuite( TableauTuples(level=6, size=1) ).run()
      sage: TestSuite( TableauTuples(level=6, size=10) ).run()

  Check that trac:\(14145\) has been fixed:

      sage: 1 in TableauTuples()
      False

  `Element`

    alias of `TableauTuple`

  `global_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.global_options(display="array")
        sage: T
          1  2  3
          4  5
        sage: Tableaux.global_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.global_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.global_options(ascii_art="normal")
        sage: ascii_art(t)
        +---+---+
        | 4 | 5 |
        +---+---+---+
        | 1 | 2 | 3 |
        +---+---+---+
        sage: Tableaux.global_options(ascii_art="compact")
        sage: ascii_art(t)
        |4|5|
        |1|2|3|
        sage: Tableaux.global_options.reset()

    See `GlobalOptions` for more features of these options.

  `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 `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]])]

  `size`()

    Return the `size` of a tableau tuple in `self`, or `None` if different tableau tuples in `self` can have sifferent 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()
        ([], [], [])

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