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# Strong and weak tableaux

There are two types of \(k\)-tableaux: strong \(k\)-tableaux and weak \(k\)-tableaux. Standard weak \(k\)-tableaux correspond to saturated chains in the weak order, whereas standard strong \(k\)-tableaux correspond to saturated chains in the strong Bruhat order. For semistandard tableaux, the notion of weak and strong horizontal strip is necessary. More information can be found in [LLMS2006] .

> See also

> `sage.combinat.k_tableau.StrongTableau()`, `sage.combinat.k_tableau.WeakTableau()`

REFERENCES:

[LLMS2006] | (1, 2) T. Lam, L. Lapointe, J. Morse, M. Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Memoirs of the AMS, 208 (2010), no. 977, Arxiv math.CO/0609110

[LLMSSZ2013] | (1, 2, 3, 4, 5, 6) T. Lam, L. Lapointe, J. Morse, A. Schilling, M. Shimozono, M. Zabrocki, \(k\)-Schur functions and affine Schubert calculus, preprint Arxiv 1301.3569

Authors:

- Anne Schilling and Mike Zabrocki (2013): initial version
- Avi Dalal and Nate Gallup (2013): implementation of \(k\)-charge

class `sage.combinat.k_tableau.``StrongTableau`(parent, T)

  Bases: `sage.structure.list_clone.ClonableList`

  A (standard) strong \(k\)-tableau is a (saturated) chain in Bruhat order.

  Combinatorially, it is a sequence of embedded \(k+1\)-cores (subject to some conditions) together with a set of markings.

  A strong cover in terms of cores corresponds to certain translated ribbons. A marking corresponds to the choice of one of the translated ribbons, which is indicated by marking the head (southeast most cell in French notation) of the chosen ribbon. For more information, see [LLMS2006] and [LLMSSZ2013].

  In Sage, a strong \(k\)-tableau is created by specifying \(k\), a standard strong tableau together with its markings, and a weight \(\mu\). Here the standard tableau is represented by a sequence of \(k+1\)-cores

  \[\lambda^{(0)} \subseteq \lambda^{(1)} \subseteq \cdots \subseteq \lambda^{(m)}\]

  where each of the \(\lambda^{(i)}\) is a \(k+1\)-core. The standard tableau is a filling of the diagram for the core \(\lambda^{(m)}/\lambda^{(0)}\) where a strong cover is represented by letters \(\pm i\) in the skew shape \(\lambda^{(i)}/\lambda^{(i-1)}\). Each skew \((k+1)\)-core \(\lambda^{(i)}/\lambda^{(i-1)}\) is a ribbon or multiple copies of the same ribbon which are separated by \(k+1\) diagonals. Precisely one of the copies of the ribbons will be marked in the largest diagonal of the connected component (the ‘head’ of the ribbon). The marked cells are indicated by negative signs.

  The strong tableau is stored as a standard strong marked tableau (referred to as the standard part of the strong tableau) and a vector representing the weight.

  EXAMPLES:

      sage: StrongTableau( [[-1, -2, -3], [3]], 2, [3] )
      [[-1, -1, -1], [1]]
      sage: StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1])
      [[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4]]

  Alternatively, the strong \(k\)-tableau can also be entered directly in semistandard format and then the standard tableau and the weight are computed and stored:

      sage: T = StrongTableau([[-1,-1,-1],[1]], 2); T
      [[-1, -1, -1], [1]]
      sage: T.to_standard_list()
      [[-1, -2, -3], [3]]
      sage: T.weight()
      (3,)
      sage: T = StrongTableau([[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4]], 3); T
      [[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4]]
      sage: T.to_standard_list()
      [[-1, -2, -4, -7], [-3, 6, -6, 8], [4, 7], [-5, -8]]
      sage: T.weight()
      (2, 2, 3, 1)

  `cell_of_highest_head`(v)

    Return the cell of the highest head of label `v` in the standard part of `self`.

    Return the cell where the head of the ribbon in the highest row is located in the underlying standard tableau. If there is no cell with entry `v` then the cell returned is \((0, r)\) where \(r\) is the length of the first row.

    This cell is calculated by iterating through the diagonals of the tableau.

    INPUT:

    - `v` – an integer indicating the label in the standard tableau

    OUTPUT:

    - a pair of integers indicating the coordinates of the head of the highest ribbon with label `v`

    EXAMPLES:

        sage: T = StrongTableau([[-1,2,-3],[-2,3],[3]], 1)
        sage: [T.cell_of_highest_head(v) for v in range(1,5)]
        [(0, 0), (1, 0), (2, 0), (0, 3)]
        sage: T = StrongTableau([[None,None,-3,4],[3,-4]],2)
        sage: [T.cell_of_highest_head(v) for v in range(1,5)]
        [(1, 0), (1, 1), (0, 4), (0, 4)]

    TESTS:

        sage: StrongTableau([],2).cell_of_highest_head(1)
        (0, 0)

  `cell_of_marked_head`(v)

    Return location of marked head labeled by `v` in the standard part of `self`.

    Return the coordinates of the `v`-th marked cell in the strong standard tableau `self`. If there is no mark, then the value returned is \((0, r)\) where \(r\) is the length of the first row.

    INPUT:

    - `v` – an integer representing the label in the standard tableau

    OUTPUT:

    - a pair of the coordinates of the marked cell with entry `v`

    EXAMPLES:

        sage: T = StrongTableau([[-1, -3, 4, -5], [-2], [-4]], 3)
        sage: [ T.cell_of_marked_head(i) for i in range(1,7)]
        [(0, 0), (1, 0), (0, 1), (2, 0), (0, 3), (0, 4)]
        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: [ T.cell_of_marked_head(i) for i in range(1,7)]
        [(2, 0), (0, 2), (2, 1), (0, 3), (4, 0), (0, 4)]

    TESTS:

        sage: StrongTableau([],4).cell_of_marked_head(4)
        (0, 0)

  `cells_head_dictionary`()

    Return a dictionary with the locations of the heads of all markings.

    Return a dictionary of values and lists of cells where the heads with the values are located.

    OUPUT:

    - a dictionary with keys the entries in the tableau and values are the coordinates of the heads with those entries

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,-4,7],[-3,6,-6,8],[4,-7],[-5,-8]], 3)
        sage: T.cells_head_dictionary()
        {1: [(0, 0)],
         2: [(0, 1)],
         3: [(1, 0)],
         4: [(2, 0), (0, 2)],
         5: [(3, 0)],
         6: [(1, 2)],
         7: [(2, 1), (0, 3)],
         8: [(3, 1), (1, 3)]}
        sage: T = StrongTableau([[None, 4, -4, -6, -7, 8, 8, -8], [None, -5, 8, 8, 8], [-3, 6]],3)
        sage: T.cells_head_dictionary()
        {1: [(2, 0)],
         2: [(0, 2)],
         3: [(1, 1)],
         4: [(2, 1), (0, 3)],
         5: [(0, 4)],
         6: [(1, 4), (0, 7)]}
         sage: StrongTableau([[None, None], [None, -1]], 4).cells_head_dictionary()
         {1: [(1, 1)]}

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).cells_head_dictionary()
        {}
        sage: StrongTableau([],4).cells_head_dictionary()
        {}

  `cells_of_heads`(v)

    Return a list of cells of the heads with label `v` in the standard part of `self`.

    A list of cells which are heads of the ribbons with label `v` in the standard part of the tableau `self`. If there is no cell labelled by `v` then return the empty list.

    INPUT:

    - `v` – an integer label

    OUPUT:

    - a list of pairs of integers of the coordinates of the heads of the ribbons with label `v`

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.cells_of_heads(1)
        [(2, 0)]
        sage: T.cells_of_heads(2)
        [(3, 0), (0, 2)]
        sage: T.cells_of_heads(3)
        [(2, 1)]
        sage: T.cells_of_heads(4)
        [(3, 1), (0, 3)]
        sage: T.cells_of_heads(5)
        [(4, 0)]
        sage: T.cells_of_heads(6)
        []

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).cells_of_heads(1)
        []
        sage: StrongTableau([],4).cells_of_heads(1)
        []

  `cells_of_marked_ribbon`(v)

    Return a list of all cells the marked ribbon labeled by `v` in the standard part of `self`.

    Return the list of coordinates of the cells which are in the marked ribbon with label `v` in the standard part of the tableau. Note that the result is independent of the weight of the tableau.

    The cells are listed from largest content (where the mark is located) to the smallest. Hence, the first entry in this list will be the marked cell.

    INPUT:

    - `v` – the entry of the standard tableau

    OUTPUT:

    - a list of pairs representing the coordinates of the cells of the marked ribbon

    EXAMPLES:

        sage: T = StrongTableau([[-1, -1, -2, -2, 3], [2, -3], [-3]],3)
        sage: T.to_standard_list()
        [[-1, -2, -3, -4, 6], [4, -6], [-5]]
        sage: T.cells_of_marked_ribbon(1)
        [(0, 0)]
        sage: T.cells_of_marked_ribbon(4)
        [(0, 3)]
        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3)
        sage: T.cells_of_marked_ribbon(6)
        [(1, 2), (1, 1)]
        sage: T.cells_of_marked_ribbon(9)
        []
        sage: T = StrongTableau([[None, None, -1, -1, 3], [1, -3], [-3]],3)
        sage: T.to_standard_list()
        [[None, None, -1, -2, 4], [2, -4], [-3]]
        sage: T.cells_of_marked_ribbon(1)
        [(0, 2)]

    TESTS:

        sage: StrongTableau([],3).cells_of_marked_ribbon(1)
        []

  `check`()

    Check that `self` is a valid strong \(k\)-tableau.

    This function verifies that the outer and inner shape of the parent class is equal to the outer and inner shape of the tableau, that the tableau portion of `self` is a valid standard tableau, that the marks are placed correctly and that the size and weight agree.

    EXAMPLES:

        sage: T = StrongTableau([[-1, -1, -2], [2]], 2)
        sage: T.check()
        sage: T = StrongTableau([[None, None, 2, -4, -4], [-1, 4], [-2]], 3)
        sage: T.check()

    TESTS:

        sage: ST = StrongTableaux(2, [3,1], [1,1,1,1])
        sage: ST([[-1,-2,3],[-3]])
        Traceback (most recent call last):
        ...
        ValueError: The size of the tableau [[-1, -2, 3], [-3]] and weight (1, 1, 1, 1) do not match
        sage: ST([[-1,-3],[-2],[3]])
        Traceback (most recent call last):
        ...
        ValueError: The outer shape of the parent does not agree with the outer shape of the tableau!

        sage: StrongTableau([[-1, -2, 2], [1]], 2)
        Traceback (most recent call last):
        ...
        ValueError: The marks in [[-1, -2, 2], [1]] are not correctly placed.

        sage: StrongTableau([[-1, -2, 3], [3]], 2)
        Traceback (most recent call last):
        ...
        ValueError: The marks in [[-1, -2, 3], [3]] are not correctly placed.

        sage: StrongTableau([[-1,-2,-4,7],[-3,6,-6,8],[4,-7],[-5,-8]], 3, [2,2,3,1])
        Traceback (most recent call last):
        ...
        ValueError: The weight=(2, 2, 3, 1) and the markings on the standard tableau=[[-1, -2, -4, 7], [-3, 6, -6, 8], [4, -7], [-5, -8]] do not agree.

  `content_of_highest_head`(v)

    Return the diagonal of the highest head of the cells labeled `v` in the standard part of `self`.

    Return the content of the cell of the head in the highest row of all ribbons labeled by `v` of the underlying standard tableau. If there is no cell with entry `v` then the value returned is the length of the first row.

    INPUT:

    - `v` – an integer representing the label in the standard tableau

    OUTPUT:

    - an integer representing the content of the head of the highest ribbon with label `v`

    EXAMPLES:

        sage: [StrongTableau([[-1,2,-3],[-2,3],[3]], 1).content_of_highest_head(v) for v in range(1,5)]
        [0, -1, -2, 3]

    TESTS:

        sage: StrongTableau([], 4).content_of_highest_head(1)
        0
        sage: StrongTableau([[-1,-1]], 4).content_of_highest_head(3)
        2

  `content_of_marked_head`(v)

    Return the diagonal of the marked label `v` in the standard part of `self`.

    Return the content (the \(j-i\) coordinate of the cell) of the `v`-th marked cell in the strong standard tableau `self`. If there is no mark, then the value returned is the size of first row.

    INPUT:

    - `v` – an integer representing the label in the standard tableau

    OUTPUT:

    - an integer representing the residue of the location of the mark

    EXAMPLES:

        sage: [ StrongTableau([[-1, -3, 4, -5], [-2], [-4]], 3).content_of_marked_head(i) for i in range(1,7)]
        [0, -1, 1, -2, 3, 4]
        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: [ T.content_of_marked_head(i) for i in range(1,7)]
        [-2, 2, -1, 3, -4, 4]

    TESTS:

        sage: StrongTableau([],4).content_of_marked_head(4)
        0

  `contents_of_heads`(v)

    A list of contents of the cells which are heads of the ribbons with label `v`.

    If there is no cell labelled by `v` then return the empty list.

    INPUT:

    - `v` – an integer label

    OUPUT:

    - a list of integers of the content of the heads of the ribbons with label `v`

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.contents_of_heads(1)
        [-2]
        sage: T.contents_of_heads(2)
        [-3, 2]
        sage: T.contents_of_heads(3)
        [-1]
        sage: T.contents_of_heads(4)
        [-2, 3]
        sage: T.contents_of_heads(5)
        [-4]
        sage: T.contents_of_heads(6)
        []

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).contents_of_heads(1)
        []
        sage: StrongTableau([],4).contents_of_heads(1)
        []

  `entries_by_content`(diag)

    Return the entries on the diagonal of `self`.

    Return the entries in the tableau that are in the cells \((i,j)\) with \(j-i\) equal to `diag` (that is, with content equal to `diag`).

    INPUT:

    - `diag` – an integer indicating the diagonal

    OUTPUT:

    - a list (perhaps empty) of labels on the diagonal `diag`

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.entries_by_content(0)
        []
        sage: T.entries_by_content(1)
        []
        sage: T.entries_by_content(2)
        [-1]
        sage: T.entries_by_content(-2)
        [-1, 2]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).entries_by_content(1)
        []
        sage: StrongTableau([],4).entries_by_content(1)
        []

  `entries_by_content_standard`(diag)

    Return the entries on the diagonal of the standard part of `self`.

    Return the entries in the tableau that are in the cells \((i,j)\) with \(j-i\) equal to `diag` (that is, with content equal to `diag`) in the standard tableau.

    INPUT:

    - `diag` – an integer indicating the diagonal

    OUTPUT:

    - a list (perhaps empty) of labels on the diagonal `diag`

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.entries_by_content_standard(0)
        []
        sage: T.entries_by_content_standard(1)
        []
        sage: T.entries_by_content_standard(2)
        [-2]
        sage: T.entries_by_content_standard(-2)
        [-1, 4]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).entries_by_content_standard(1)
        []
        sage: StrongTableau([],4).entries_by_content_standard(1)
        []

  `follows_tableau`()

    Return a list of strong marked tableaux with length one longer than `self`.

    Return list of all strong tableaux obtained from `self` by extending to a core which follows the shape of `self` in the strong order.

    OUTPUT:

    - a list of strong tableaux which follow `self` in strong order

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1])
        sage: T.follows_tableau()
        [[[-1, -1, -2, -3, 5, 5, -5], [-2, 3, -3, 4], [2, 3], [-3, -4]],
         [[-1, -1, -2, -3, 5], [-2, 3, -3, 4], [2, 3, 5], [-3, -4], [-5]],
         [[-1, -1, -2, -3, 5], [-2, 3, -3, 4], [2, 3, -5], [-3, -4], [5]],
         [[-1, -1, -2, -3, -5], [-2, 3, -3, 4], [2, 3, 5], [-3, -4], [5]],
         [[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4], [-5], [5], [5]]]
        sage: StrongTableau([[-1,-2],[-3,-4]],3).follows_tableau()
        [[[-1, -2, 5, 5, -5], [-3, -4]], [[-1, -2, 5], [-3, -4], [-5]],
         [[-1, -2, -5], [-3, -4], [5]], [[-1, -2], [-3, -4], [-5], [5], [5]]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).follows_tableau()
        [[[None, None, -1], [None]], [[None, None], [None, -1]], [[None, None], [None], [-1]]]
        sage: StrongTableau([],4).follows_tableau()
        [[[-1]]]

  `height_of_ribbon`(v)

    The number of rows occupied by one of the ribbons with label `v`.

    The number of rows occupied by the marked ribbon with label `v` (and by consequence the number of rows occupied by any ribbon with the same label) in the standard part of `self`.

    INPUT:

    - `v` – the label of the standard marked tableau

    OUTPUT:

    - a non-negative integer representing the number of rows occupied by the ribbon which is marked

    EXAMPLES:

        sage: T = StrongTableau([[-1, -1, -2, -2, 3], [2, -3], [-3]],3)
        sage: T.to_standard_list()
        [[-1, -2, -3, -4, 6], [4, -6], [-5]]
        sage: T.height_of_ribbon(1)
        1
        sage: T.height_of_ribbon(4)
        1
        sage: T = StrongTableau([[None,None,1,-2],[None,-3,4,-5],[-1,3],[-4,5]], 3)
        sage: T.height_of_ribbon(3)
        2
        sage: T.height_of_ribbon(6)
        0

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).height_of_ribbon(1)
        0
        sage: StrongTableau([],4).height_of_ribbon(1)
        0

  `inner_shape`()

    Return the inner shape of `self`.

    If `self` is a strong skew tableau, then this method returns the inner shape (the shape of the cells labelled with `None`). If `self` is not skew, then the inner shape is empty.

    OUTPUT:

    - a \((k+1)\)-core

    EXAMPLES:

        sage: StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).inner_shape()
        [2, 2]
        sage: StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1]).inner_shape()
        []

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).inner_shape()
        [2, 1]
        sage: StrongTableau([],4).inner_shape()
        []

  `intermediate_shapes`()

    Return the intermediate shapes of `self`.

    A (skew) tableau with letters \(1, 2, \ldots, \ell\) can be viewed as a sequence of shapes, where the \(i\)-th shape is given by the shape of the subtableau on letters \(1, 2, \ldots, i\).

    The output is the list of these shapes. The marked cells are ignored so to recover the strong tableau one would need the intermediate shapes and the `content_of_marked_head()` for each pair of adjacent shapes in the list.

    OUTPUT:

    - a list of lists of integers representing \(k+1\)-cores

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1])
        sage: T.intermediate_shapes()
        [[], [2], [3, 1, 1], [4, 3, 2, 1], [4, 4, 2, 2]]
        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.intermediate_shapes()
        [[2, 2], [3, 2, 1, 1], [4, 2, 2, 2], [4, 2, 2, 2, 1, 1, 1, 1]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).intermediate_shapes()
        [[2, 1]]
        sage: StrongTableau([],4).intermediate_shapes()
        [[]]

  `is_column_strict_with_weight`(mu)

    Test if `self` is a column strict tableau with respect to the weight `mu`.

    INPUT:

    - `mu` – a vector of weights

    OUTPUT:

    - a boolean, `True` means the underlying column strict strong marked tableau is valid

    EXAMPLES:

        sage: StrongTableau([[-1, -2, -3], [3]], 2).is_column_strict_with_weight([3])
        True
        sage: StrongTableau([[-1, -2, 3], [-3]], 2).is_column_strict_with_weight([3])
        False

    TESTS:

        sage: StrongTableau([[None, None, None], [None]], 2).is_column_strict_with_weight([])
        True
        sage: StrongTableau([], 4).is_column_strict_with_weight([])
        True

  `left_action`(tij)

    Action of transposition `tij` on `self` by adding marked ribbons.

    Computes the left action of the transposition `tij` on the tableau. If `tij` acting on the element of the affine grassmannian raises the length by 1, then this function will add a cell to the standard tableau.

    INPUT:

    - `tij` – a transposition represented as a pair \((i, j)\).

    OUPUT:

    - `self` after it has been modified by the action of the transposition `tij`

    EXAMPLES:

        sage: StrongTableau( [[None, -1, -2, -3], [3], [-4]], 3, weight=[1,1,1,1] ).left_action([0,1])
        [[None, -1, -2, -3, 5], [3, -5], [-4]]
        sage: StrongTableau( [[None, -1, -2, -3], [3], [-4]], 3, weight=[1,1,1,1] ).left_action([4,5])
        [[None, -1, -2, -3, -5], [3, 5], [-4]]
        sage: T = StrongTableau( [[None, -1, -2, -3], [3], [-4]], 3, weight=[1,1,1,1] )
        sage: T.left_action([-3,-2])
        [[None, -1, -2, -3], [3], [-4], [-5]]
        sage: T = StrongTableau( [[None, -1, -2, -3], [3], [-4]], 3, weight=[3,1] )
        sage: T.left_action([-3,-2])
        [[None, -1, -1, -1], [1], [-2], [-3]]
        sage: T
        [[None, -1, -1, -1], [1], [-2]]
        sage: T.check()
        sage: T.weight()
        (3, 1)

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).left_action([-2,-1])
        [[None, None], [None], [-1]]
        sage: StrongTableau([],4).left_action([0,1])
        [[-1]]

  `number_of_connected_components`(v)

    Number of connected components of ribbons with label `v` in the standard part.

    The number of connected components is calculated by finding the number of cells with label `v` in the standard part of the tableau and dividing by the number of cells in the ribbon.

    INPUT:

    - `v` – the label of the standard marked tableau

    OUTPUT:

    - a non-negative integer representing the number of connected components

    EXAMPLES:

        sage: T = StrongTableau([[-1, -1, -2, -2, 3], [2, -3], [-3]],3)
        sage: T.to_standard_list()
        [[-1, -2, -3, -4, 6], [4, -6], [-5]]
        sage: T.number_of_connected_components(1)
        1
        sage: T.number_of_connected_components(4)
        2
        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3)
        sage: T.number_of_connected_components(6)
        1
        sage: T.number_of_connected_components(9)
        0

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).number_of_connected_components(1)
        0
        sage: StrongTableau([],4).number_of_connected_components(1)
        0

  `outer_shape`()

    Return the outer shape of `self`.

    This method returns the outer shape of `self` as viewed as a `Core`. The outer shape of a strong tableau is always a \((k+1)\)-core.

    OUTPUT:

    - a \((k+1)\)-core

    EXAMPLES:

        sage: StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).outer_shape()
        [4, 2, 2, 2, 1, 1, 1, 1]
        sage: StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1]).outer_shape()
        [4, 4, 2, 2]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).outer_shape()
        [2, 1]
        sage: StrongTableau([],4).outer_shape()
        []

  `pp`()

    Print the strong tableau `self` in pretty print format.

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1])
        sage: T.pp()
        -1 -1 -2 -3
        -2  3 -3  4
         2  3
        -3 -4
        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.pp()
          .  . -1 -2
          .  .
         -1 -2
          1  2
         -3
          3
          3
          3
        sage: Tableaux.global_options(convention="French")
        sage: T.pp()
          3
          3
          3
         -3
          1  2
         -1 -2
          .  .
          .  . -1 -2
        sage: Tableaux.global_options(convention="English")

  `restrict`(r)

    Restrict the standard part of the tableau to the labels \(1, 2, \ldots, r\).

    Return the tableau consisting of the labels of the standard part of `self` restricted to the labels of \(1\) through `r`. The result is another `StrongTableau` object.

    INPUT:

    - `r` – an integer

    OUTPUT:

    - A strong tableau

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -4, 5, -5], [None, None], [-1, -3], [-2], [2], [2], [3]], 4, weight=[1,1,1,1,1])
        sage: T.restrict(3)
        [[None, None], [None, None], [-1, -3], [-2], [2], [2], [3]]
        sage: TT = T.restrict(0)
        sage: TT
        [[None, None], [None, None]]
        sage: TT == StrongTableau( [[None, None], [None, None]], 4 )
        True
        sage: T.restrict(5) == T
        True

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).restrict(1)
        [[None, None], [None]]
        sage: StrongTableau([],4).restrict(1)
        []

  `ribbons_above_marked`(v)

    Number of ribbons of label `v` higher than the marked ribbon in the standard part.

    Return the number of copies of the ribbon with label `v` in the standard part of `self` which are in a higher row than the marked ribbon. Note that the result is independent of the weight of the tableau.

    INPUT:

    - `v` – the entry of the standard tableau

    OUTPUT:

    - an integer representing the number of copies of the ribbon above the marked ribbon

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3)
        sage: T.ribbons_above_marked(4)
        1
        sage: T.ribbons_above_marked(6)
        0
        sage: T.ribbons_above_marked(9)
        0
        sage: StrongTableau([[-1,-2,-3,-4],[2,3,4],[3,4],[4]], 1).ribbons_above_marked(4)
        3

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).ribbons_above_marked(1)
        0
        sage: StrongTableau([],4).ribbons_above_marked(1)
        0

  `set_weight`(mu)

    Sets a new weight `mu` for `self`.

    This method first tests if the underlying standard tableau is column-strict with respect to the weight `mu`. If it is, then it changes the weight and returns the tableau; otherwise it raises an error.

    INPUT:

    - `mu` – a list of non-negative integers representing the new weight

    EXAMPLES:

        sage: StrongTableau( [[-1, -2, -3], [3]], 2 ).set_weight( [3] )
        [[-1, -1, -1], [1]]
        sage: StrongTableau( [[-1, -2, -3], [3]], 2 ).set_weight( [0,3] )
        [[-2, -2, -2], [2]]
        sage: StrongTableau( [[-1, -2, 3], [-3]], 2 ).set_weight( [2, 0, 1] )
        [[-1, -1, 3], [-3]]
        sage: StrongTableau( [[-1, -2, 3], [-3]], 2 ).set_weight( [3] )
        Traceback (most recent call last):
        ...
        ValueError: [[-1, -2, 3], [-3]] is not a semistandard strong tableau with respect to the partition [3]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).set_weight([])
        [[None, None], [None]]
        sage: StrongTableau([],4).set_weight([])
        []

  `shape`()

    Return the shape of `self`.

    If `self` is a skew tableau then return a pair of \(k+1\)-cores consisting of the outer and the inner shape. If `self` is strong tableau with no inner shape then return a \(k+1\)-core.

    INPUT:

    - `form` - optional argument to indicate ‘inner’, ‘outer’ or ‘skew’ (default : ‘outer’)

    OUTPUT:

    - a \(k+1\)-core or a pair of \(k+1\)-cores if form is not ‘inner’ or ‘outer’

    EXAMPLES:

        sage: T = StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4)
        sage: T.shape()
        ([4, 2, 2, 2, 1, 1, 1, 1], [2, 2])
        sage: StrongTableau([[-1, -2, 3], [-3]], 2).shape()
        [3, 1]
        sage: type(StrongTableau([[-1, -2, 3], [-3]], 2).shape())
        <class 'sage.combinat.core.Cores_length_with_category.element_class'>

    TESTS:

        sage: StrongTableau([[None, None, None], [None]], 2).shape()
        ([3, 1], [3, 1])
        sage: StrongTableau([],4).shape()
        []

  `size`()

    Return the size of the strong tableau.

    The size of the strong tableau is the sum of the entries in the `weight()`. It will also be equal to the length of the outer shape (as a \(k+1\)-core) minus the length of the inner shape.

    See also

    `sage.combinat.core.Core.length()`

    OUTPUT:

    - a non-negative integer

    EXAMPLES:

        sage: StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3).size()
        5
        sage: StrongTableau([[None, None, -1, 2], [-2], [-3]], 3).size()
        3

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).size()
        0
        sage: StrongTableau([],4).size()
        0

  `spin`()

    Return the spin statistic of the tableau `self`.

    The spin is an integer statistic on a strong marked tableau. It is the sum of \((h-1) r\) plus the number of connected components above the marked one where \(h\) is the height of the marked ribbon and \(r\) is the number of connected components.

    See also

    `height_of_ribbon()`, `number_of_connected_components()`, `ribbons_above_marked()`

    The \(k\)-Schur functions with a parameter \(t\) can be defined as

    \[s^{(k)}_\lambda[X; t] = \sum_T t^{spin(T)} m_{weight(T)}[X]\]

    where the sum is over all column strict marked strong \(k\)-tableaux of shape \(\lambda\) and partition content.

    OUTPUT:

    - an integer value representing the spin.

    EXAMPLES:

        sage: StrongTableau([[-1,-2,5,6],[-3,-4,-7,8],[-5,-6],[7,-8]], 3, [2,2,3,1]).spin()
        1
        sage: StrongTableau([[-1,-2,-4,-7],[-3,6,-6,8],[4,7],[-5,-8]], 3, [2,2,3,1]).spin()
        2
        sage: StrongTableau([[None,None,-1,-3],[-2,3,-3,4],[2,3],[-3,-4]], 3).spin()
        2
        sage: ks3 = SymmetricFunctions(QQ['t'].fraction_field()).kschur(3)
        sage: t = ks3.realization_of().t
        sage: m = ks3.ambient().realization_of().m()
        sage: myks221 = sum(sum(t**T.spin() for T in StrongTableaux(3,[3,2,1],weight=mu))*m(mu) for mu in Partitions(5, max_part=3))
        sage: myks221 == m(ks3[2,2,1])
        True
        sage: h = ks3.ambient().realization_of().h()
        sage: Core([4,4,2,2],4).to_bounded_partition()
        [2, 2, 2, 2]
        sage: ks3[2,2,2,2].lift().scalar(h[3,3,2]) == sum( t**T.spin() for T in StrongTableaux(3, [4,4,2,2], weight=[3,3,2]) )
        True

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).spin()
        0
        sage: StrongTableau([],4).spin()
        0

  `spin_of_ribbon`(v)

    Return the spin of the ribbon with label `v` in the standard part of `self`.

    The spin of a ribbon is an integer statistic. It is the sum of \((h-1) r\) plus the number of connected components above the marked one where \(h\) is the height of the marked ribbon and \(r\) is the number of connected components.

    See also

    `height_of_ribbon()`, `number_of_connected_components()`, `ribbons_above_marked()`

    INPUT:

    - `v` – a label of the standard part of the tableau

    OUTPUT:

    - an integer value representing the spin of the ribbon with label `v`.

    EXAMPLES:

        sage: T = StrongTableau([[-1,-2,5,6],[-3,-4,-7,8],[-5,-6],[7,-8]], 3)
        sage: [T.spin_of_ribbon(v) for v in range(1,9)]
        [0, 0, 0, 0, 0, 0, 1, 0]
        sage: T = StrongTableau([[None,None,-1,-3],[-2,3,-3,4],[2,3],[-3,-4]], 3)
        sage: [T.spin_of_ribbon(v) for v in range(1,7)]
        [0, 1, 0, 0, 1, 0]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).spin_of_ribbon(1)
        0
        sage: StrongTableau([],4).spin_of_ribbon(1)
        0

  `to_list`()

    Return the marked column strict (possibly skew) tableau as a list of lists.

    OUTPUT:

    - a list of lists of integers or `None`

    EXAMPLES:

        sage: StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3).set_weight([2,1,1,1]).to_list()
        [[-1, -1, -2, 3], [-3], [-4]]
        sage: StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).to_list()
        [[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]]
        sage: StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3, [3,1,1]).to_list()
        [[-1, -1, -1, 2], [-2], [-3]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_list()
        [[None, None], [None]]
        sage: StrongTableau([],4).to_list()
        []

  `to_standard_list`()

    Return the underlying standard strong tableau as a list of lists.

    Internally, for a strong tableau the standard strong tableau and its weight is stored separately. This method returns the underlying standard part.

    OUTPUT:

    - a list of lists of integers or `None`

    EXAMPLES:

        sage: StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3, [3,1,1]).to_standard_list()
        [[-1, -2, -3, 4], [-4], [-5]]
        sage: StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).to_standard_list()
        [[None, None, -2, -4], [None, None], [-1, -3], [2, 4], [-5], [5], [5], [5]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_standard_list()
        [[None, None], [None]]
        sage: StrongTableau([],4).to_standard_list()
        []

  `to_standard_tableau`()

    Return the underlying standard strong tableau as a `StrongTableau` object.

    Internally, for a strong tableau the standard strong tableau and its weight is stored separately. This method returns the underlying standard part as a `StrongTableau`.

    OUTPUT:

    - a strong tableau with standard weight

    EXAMPLES:

        sage: T = StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3, [3,1,1])
        sage: T.to_standard_tableau()
        [[-1, -2, -3, 4], [-4], [-5]]
        sage: T.to_standard_tableau() == T.to_standard_list()
        False
        sage: StrongTableau([[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).to_standard_tableau()
        [[None, None, -2, -4], [None, None], [-1, -3], [2, 4], [-5], [5], [5], [5]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_standard_tableau()
        [[None, None], [None]]
        sage: StrongTableau([],4).to_standard_tableau()
        []

  `to_transposition_sequence`()

    Return a list of transpositions corresponding to `self`.

    Given a strong column strict tableau `self` returns the list of transpositions which when applied to the left of an empty tableau gives the corresponding strong standard tableau.

    OUTPUT:

    - a list of pairs of values `[i,j]` representing the transpositions \(t_{ij}\)

    EXAMPLES:

        sage: T = StrongTableau([[-1, -1, -1], [1]],2)
        sage: T.to_transposition_sequence()
        [[2, 3], [1, 2], [0, 1]]
        sage: T = StrongTableau([[-1, -1, 2], [-2]],2)
        sage: T.to_transposition_sequence()
        [[-1, 0], [1, 2], [0, 1]]
        sage: T = StrongTableau([[None, -1, 2, -3], [-2, 3]],2)
        sage: T.to_transposition_sequence()
        [[3, 4], [-1, 0], [1, 2]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_transposition_sequence()
        []
        sage: StrongTableau([],4).to_transposition_sequence()
        []

  `to_unmarked_list`()

    Return the tableau as a list of lists with markings removed.

    Return the list of lists of the rows of the tableau where the markings have been removed.

    OUTPUT:

    - a list of lists of integers or `None`

    EXAMPLES:

        sage: T = StrongTableau( [[-1, -2, -3, 4], [-4], [-5]], 3, [3,1,1])
        sage: T.to_unmarked_list()
        [[1, 1, 1, 2], [2], [3]]
        sage: TT = T.set_weight([2,1,1,1])
        sage: TT.to_unmarked_list()
        [[1, 1, 2, 3], [3], [4]]
        sage: StrongTableau( [[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).to_unmarked_list()
        [[None, None, 1, 2], [None, None], [1, 2], [1, 2], [3], [3], [3], [3]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_unmarked_list()
        [[None, None], [None]]
        sage: StrongTableau([],4).to_unmarked_list()
        []

  `to_unmarked_standard_list`()

    Return the standard part of the tableau as a list of lists with markings removed.

    Return the list of lists of the rows of the tableau where the markings have been removed.

    OUTPUT:

    - a list of lists of integers or `None`

    EXAMPLES:

        sage: StrongTableau( [[-1, -2, -3, 4], [-4], [-5]], 3, [3,1,1]).to_unmarked_standard_list()
        [[1, 2, 3, 4], [4], [5]]
        sage: StrongTableau( [[None, None, -1, -2], [None, None], [-1, -2], [1, 2], [-3], [3], [3], [3]], 4).to_unmarked_standard_list()
        [[None, None, 2, 4], [None, None], [1, 3], [2, 4], [5], [5], [5], [5]]

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).to_unmarked_standard_list()
        [[None, None], [None]]
        sage: StrongTableau([],4).to_unmarked_standard_list()
        []

  `weight`()

    Return the weight of the tableau.

    The weight is a list of non-negative integers indicating the number of 1s, number of 2s, number of 3s, etc.

    OUTPUT:

    - a list of non-negative integers

    EXAMPLES:

        sage: T = StrongTableau([[-1, -2, -3, 4], [-4], [-5]], 3); T.weight()
        (1, 1, 1, 1, 1)
        sage: T.set_weight([3,1,1]).weight()
        (3, 1, 1)
        sage: StrongTableau([[-1,-1,-2,-3],[-2,3,-3,4],[2,3],[-3,-4]], 3).weight()
        (2, 2, 3, 1)

    TESTS:

        sage: StrongTableau([[None, None], [None]], 4).weight()
        ()
        sage: StrongTableau([],4).weight()
        ()

class `sage.combinat.k_tableau.``StrongTableaux`(k, shape, weight)

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

  TESTS:

      sage: strongT = StrongTableaux(2, [3,1], weight=[2,1])
      sage: TestSuite(strongT).run()

      sage: strongT = StrongTableaux(0, [2,2], weight=[2,2])
      Traceback (most recent call last):
      ...
      ValueError: The input k has to be a positive integer

  `Element`

    alias of `StrongTableau`

  classmethod `add_marking`(unmarkedT, marking, k, weight)

    Add markings to a partially marked strong tableau.

    Given an partially marked standard tableau and a list of cells where the marks should be placed along with a `weight`, return the semi-standard marked strong tableau. The marking should complete the marking so that the result is a strong standard marked tableau.

    INPUT:

    - `unmarkedT` - a list of lists which is a partially marked strong \(k\)-tableau
    - `marking` - a list of pairs of coordinates where cells are to be marked
    - `k` - a positive integer
    - `weight` - a tuple of the weight of the output tableau

    OUTPUT:

    - a `StrongTableau` object

    EXAMPLES:

        sage: StrongTableaux.add_marking([[None,1,2],[2]], [(0,1), (1,0)], 2, [1,1])
        [[None, -1, 2], [-2]]
        sage: StrongTableaux.add_marking([[None,1,2],[2]], [(0,1), (1,0)], 2, [2])
        Traceback (most recent call last):
        ...
        ValueError: The weight=(2,) and the markings on the standard tableau=[[None, -1, 2], [-2]] do not agree.
        sage: StrongTableaux.add_marking([[None,1,2],[2]], [(0,1), (0,2)], 2, [2])
        [[None, -1, -1], [1]]

    TESTS:

        sage: StrongTableaux.add_marking([[None,None,None],[None]], [], 2, [])
        [[None, None, None], [None]]
        sage: StrongTableaux.add_marking([], [], 2, [])
        []

  `an_element`()

    Return the first generated element of the class of `StrongTableaux`.

    EXAMPLES:

        sage: ST = StrongTableaux(3, [3], weight=[3])
        sage: ST.an_element()
        [[-1, -1, -1]]

  classmethod `cells_head_dictionary`(T)

    Return a dictionary with the locations of the heads of all markings.

    Return a dictionary of values and lists of cells where the heads with the values are located in a strong standard unmarked tableau `T`.

    INPUT:

    - `T` – a strong standard unmarked tableau as a list of lists

    OUPUT:

    - a dictionary with keys the entries in the tableau and values are the coordinates of the heads with those entries

    EXAMPLES:

        sage: StrongTableaux.cells_head_dictionary([[1,2,4,7],[3,6,6,8],[4,7],[5,8]])
        {1: [(0, 0)],
         2: [(0, 1)],
         3: [(1, 0)],
         4: [(2, 0), (0, 2)],
         5: [(3, 0)],
         6: [(1, 2)],
         7: [(2, 1), (0, 3)],
         8: [(3, 1), (1, 3)]}
        sage: StrongTableaux.cells_head_dictionary([[None, 2, 2, 4, 5, 6, 6, 6], [None, 3, 6, 6, 6], [1, 4]])
        {1: [(2, 0)],
         2: [(0, 2)],
         3: [(1, 1)],
         4: [(2, 1), (0, 3)],
         5: [(0, 4)],
         6: [(1, 4), (0, 7)]}

    TESTS:

        sage: StrongTableaux.cells_head_dictionary([[None, None, None],[None]])
        {}
        sage: StrongTableaux.cells_head_dictionary([])
        {}

  classmethod `follows_tableau_unsigned_standard`(Tlist, k)

    Return a list of strong tableaux one longer in length than `Tlist`.

    Return list of all standard strong tableaux obtained from `Tlist` by extending to a core which follows the shape of `Tlist` in the strong order. It does not put the markings on the last entry that it adds but it does keep the markings on all entries smaller. The objects returned are not `StrongTableau` objects (and cannot be) because the last entry will not properly marked.

    INPUT:

    - `Tlist` – a filling of a \(k+1\)-core as a list of lists
    - `k` - an integer

    OUTPUT:

    - a list of strong tableaux which follow `Tlist` in strong order

    EXAMPLES:

        sage: StrongTableaux.follows_tableau_unsigned_standard([[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4]], 3)
        [[[-1, -1, -2, -3, 5, 5, 5], [-2, 3, -3, 4], [2, 3], [-3, -4]],
         [[-1, -1, -2, -3, 5], [-2, 3, -3, 4], [2, 3, 5], [-3, -4], [5]],
         [[-1, -1, -2, -3], [-2, 3, -3, 4], [2, 3], [-3, -4], [5], [5], [5]]]
        sage: StrongTableaux.follows_tableau_unsigned_standard([[None,-1],[-2,-3]],3)
        [[[None, -1, 4, 4, 4], [-2, -3]], [[None, -1, 4], [-2, -3], [4]],
         [[None, -1], [-2, -3], [4], [4], [4]]]

    TESTS:

        sage: StrongTableaux.follows_tableau_unsigned_standard([[None, None, None], [None]], 2)
        [[[None, None, None, 1], [None, 1]], [[None, None, None], [None], [1]]]
        sage: StrongTableaux.follows_tableau_unsigned_standard([], 4)
        [[[1]]]

  `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.

  `inner_shape`()

    Return the inner shape of the class of strong tableaux.

    OUTPUT:

    - a \(k+1\)-core

    EXAMPLES:

        sage: StrongTableaux( 2, [3,1] ).inner_shape()
        []
        sage: type(StrongTableaux( 2, [3,1] ).inner_shape())
        <class 'sage.combinat.core.Cores_length_with_category.element_class'>
        sage: StrongTableaux( 4, [[2,1], [1]] ).inner_shape()
        [1]

  classmethod `marked_CST_to_transposition_sequence`(T, k)

    Return a list of transpositions corresponding to `T`.

    Given a strong column strict tableau `T` returns the list of transpositions which when applied to the left of an empty tableau gives the corresponding strong standard tableau.

    INPUT:

    - `T` – a non-empty column strict tableau as a list of lists
    - `k` – a positive integer

    OUTPUT:

    - a list of pairs of values `[i,j]` representing the transpositions \(t_{ij}\)

    EXAMPLES:

        sage: CST_to_trans = StrongTableaux.marked_CST_to_transposition_sequence
        sage: CST_to_trans([[-1, -1, -1], [1]], 2)
        [[2, 3], [1, 2], [0, 1]]
        sage: CST_to_trans([], 2)
        []
        sage: CST_to_trans([[-2, -2, -2], [2]], 2)
        [[2, 3], [1, 2], [0, 1]]
        sage: CST_to_trans([[-1, -2, -2, -2, -2], [-2, 2], [2]], 3)
        [[4, 5], [3, 4], [2, 3], [1, 2], [-1, 0], [0, 1]]
        sage: CST_to_trans([[-1, -2, -5, 5, -5, 5, -5], [-3, -4, 5, 5], [5]],3)
        [[5, 7], [3, 5], [2, 3], [0, 1], [-1, 0], [1, 2], [0, 1]]
        sage: CST_to_trans([[-1, -2, -3, 4, -7], [-4, -6], [-5, 6]],3)
        [[4, 5], [-1, 1], [-2, -1], [-1, 0], [2, 3], [1, 2], [0, 1]]

    TESTS:

        sage: StrongTableaux.marked_CST_to_transposition_sequence([[None, None, None], [None]], 2)
        []
        sage: StrongTableaux.marked_CST_to_transposition_sequence([], 4)
        []

  classmethod `marked_given_unmarked_and_weight_iterator`(unmarkedT, k, weight)

    An iterator generating strong marked tableaux from an unmarked strong tableau.

    Iterator which lists all marked tableaux of weight `weight` such that the standard unmarked part of the tableau is equal to `unmarkedT`.

    INPUT:

    - `unmarkedT` - a list of lists representing a strong unmarked tableau
    - `k` - a positive integer
    - `weight` - a list of non-negative integers indicating the weight

    OUTPUT:

    - an iterator that returns `StrongTableau` objects

    EXAMPLES:

        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[1,2,3],[3]], 2, [3])
        sage: list(ST)
        [[[-1, -1, -1], [1]]]
        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[1,2,3],[3]], 2, [0,3])
        sage: list(ST)
        [[[-2, -2, -2], [2]]]
        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[1,2,3],[3]], 2, [1,2])
        sage: list(ST)
        [[[-1, -2, -2], [2]]]
        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[1,2,3],[3]], 2, [2,1])
        sage: list(ST)
        [[[-1, -1, 2], [-2]], [[-1, -1, -2], [2]]]
        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[None, None, 1, 2, 4], [2, 4], [3]], 3, [3,1])
        sage: list(ST)
        []
        sage: ST = StrongTableaux.marked_given_unmarked_and_weight_iterator([[None, None, 1, 2, 4], [2, 4], [3]], 3, [2,2])
        sage: list(ST)
        [[[None, None, -1, -1, 2], [1, -2], [-2]],
         [[None, None, -1, -1, -2], [1, 2], [-2]]]

    TESTS:

        sage: list(StrongTableaux.marked_given_unmarked_and_weight_iterator([[None, None, None],[None]], 2, []))
        [[[None, None, None], [None]]]
        sage: list(StrongTableaux.marked_given_unmarked_and_weight_iterator([], 4, weight=[]))
        [[]]

  `outer_shape`()

    Return the outer shape of the class of strong tableaux.

    OUTPUT:

    - a \(k+1\)-core

    EXAMPLES:

        sage: StrongTableaux( 2, [3,1] ).outer_shape()
        [3, 1]
        sage: type(StrongTableaux( 2, [3,1] ).outer_shape())
        <class 'sage.combinat.core.Cores_length_with_category.element_class'>
        sage: StrongTableaux( 4, [[2,1], [1]] ).outer_shape()
        [2, 1]

  `shape`()

    Return the shape of `self`.

    If the `self` has an inner shape return a pair consisting of an inner and an outer shape. If the inner shape is empty then return only the outer shape.

    OUTPUT:

    - a \(k+1\)-core or a pair of \(k+1\)-cores

    EXAMPLES:

        sage: StrongTableaux( 2, [3,1] ).shape()
        [3, 1]
        sage: type(StrongTableaux( 2, [3,1] ).shape())
        <class 'sage.combinat.core.Cores_length_with_category.element_class'>
        sage: StrongTableaux( 4, [[2,1], [1]] ).shape()
        ([2, 1], [1])

  classmethod `standard_marked_iterator`(k, size, outer_shape=None, inner_shape=[])

    An iterator for generating standard strong marked tableaux.

    An iterator which generates all standard marked \(k\)-tableaux of a given `size` which are contained in `outer_shape` and contain the `inner_shape`. If `outer_shape` is `None` then there is no restriction on the shape of the tableaux which are created.

    INPUT:

    - `k` - a positive integer
    - `size` - a positive integer
    - `outer_shape` - a list which is a \(k+1\)-core (default: `None`)
    - `inner_shape` - a list which is a \(k+1\)-core (default: [])

    OUPUT:

    - an iterator which returns the standard marked tableaux with `size` cells and that are contained in `outer_shape` and contain `inner_shape`

    EXAMPLES:

        sage: list(StrongTableaux.standard_marked_iterator(2, 3))
        [[[-1, -2, 3], [-3]], [[-1, -2, -3], [3]], [[-1, -2], [-3], [3]], [[-1, 3, -3], [-2]], [[-1, 3], [-2], [-3]], [[-1, -3], [-2], [3]]]
        sage: list(StrongTableaux.standard_marked_iterator(2, 1, inner_shape=[1,1]))
        [[[None, 1, -1], [None]], [[None, 1], [None], [-1]], [[None, -1], [None], [1]]]
        sage: len(list(StrongTableaux.standard_marked_iterator(4,4)))
        10
        sage: len(list(StrongTableaux.standard_marked_iterator(4,6)))
        140
        sage: len(list(StrongTableaux.standard_marked_iterator(4,4, inner_shape=[2,2])))
        200
        sage: len(list(StrongTableaux.standard_marked_iterator(4,4, outer_shape=[5,2,2,1], inner_shape=[2,2])))
        24

    TESTS:

        sage: list(StrongTableaux.standard_marked_iterator(2,0,inner_shape=[3,1]))
        [[[None, None, None], [None]]]
        sage: list(StrongTableaux.standard_marked_iterator(4,0))
        [[]]

  classmethod `standard_unmarked_iterator`(k, size, outer_shape=None, inner_shape=[])

    An iterator for standard unmarked strong tableaux.

    An iterator which generates all unmarked tableaux of a given `size` which are contained in `outer_shape` and which contain the `inner_shape`.

    These are built recursively by building all standard marked strong tableaux of size `size` \(-1\) and adding all possible covers.

    If `outer_shape` is `None` then there is no restriction on the shape of the tableaux which are created.

    INPUT:

    - `k`, `size` - a positive integers
    - `outer_shape` - a list representing a \(k+1\)-core (default: `None`)
    - `inner_shape` - a list representing a \(k+1\)-core (default: [])

    OUTPUT:

    - an iterator which lists all standard strong unmarked tableaux with `size` cells and which are contained in `outer_shape` and contain `inner_shape`

    EXAMPLES:

        sage: list(StrongTableaux.standard_unmarked_iterator(2, 3))
        [[[1, 2, 3], [3]], [[1, 2], [3], [3]], [[1, 3, 3], [2]], [[1, 3], [2], [3]]]
        sage: list(StrongTableaux.standard_unmarked_iterator(2, 1, inner_shape=[1,1]))
        [[[None, 1, 1], [None]], [[None, 1], [None], [1]]]
        sage: len(list(StrongTableaux.standard_unmarked_iterator(4,4)))
        10
        sage: len(list(StrongTableaux.standard_unmarked_iterator(4,6)))
        98
        sage: len(list(StrongTableaux.standard_unmarked_iterator(4,4, inner_shape=[2,2])))
        92
        sage: len(list(StrongTableaux.standard_unmarked_iterator(4,4, outer_shape=[5,2,2,1], inner_shape=[2,2])))
        10

    TESTS:

        sage: list(StrongTableaux.standard_unmarked_iterator(2,0, outer_shape=[3,1], inner_shape=[3,1]))
        [[[None, None, None], [None]]]
        sage: list(StrongTableaux.standard_unmarked_iterator(4,0, outer_shape=[]))
        [[]]

  classmethod `transpositions_to_standard_strong`(transeq, k, emptyTableau=[])

    Return a strong tableau correponding to a sequence of transpositions.

    This method returns the action by left multiplication on the empty strong tableau by transpositions specified by `transeq`.

    INPUT:

    - `transeq` – a sequence of transpositions \(t_{ij}\) (a list of pairs).
    - `emptyTableau` – (default: `[]`) an empty list or a skew strong tableau possibly consisting of `None` entries

    OUTPUT:

    - a `StrongTableau` object

    EXAMPLES:

        sage: StrongTableaux.transpositions_to_standard_strong([[0,1]], 2)
        [[-1]]
        sage: StrongTableaux.transpositions_to_standard_strong([[-2,-1], [2,3]], 2, [[None, None]])
        [[None, None, -1], [1], [-2]]
        sage: StrongTableaux.transpositions_to_standard_strong([[2, 3], [1, 2], [0, 1]], 2)
        [[-1, -2, -3], [3]]
        sage: StrongTableaux.transpositions_to_standard_strong([[-1, 0], [1, 2], [0, 1]], 2)
        [[-1, -2, 3], [-3]]
        sage: StrongTableaux.transpositions_to_standard_strong([[3, 4], [-1, 0], [1, 2]], 2, [[None]])
        [[None, -1, 2, -3], [-2, 3]]

    TESTS:

        sage: StrongTableaux.transpositions_to_standard_strong([], 2, [[None, None, None], [None]])
        [[None, None, None], [None]]
        sage: StrongTableaux.transpositions_to_standard_strong([], 4, [])
        []

`sage.combinat.k_tableau.``WeakTableau`(t, k, inner_shape=[], representation='core')

  This is the dispatcher method for the element class of weak \(k\)-tableaux.

  Standard weak \(k\)-tableaux correspond to saturated chains in the weak order. There are three formulations of weak tableaux, one in terms of cores, one in terms of \(k\)-bounded partitions, and one in terms of factorizations of affine Grassmannian elements. For semistandard weak \(k\)-tableaux, all letters of the same value have to satisfy the conditions of a horizontal strip. In the affine Grassmannian formulation this means that all factors are cyclically decreasing elements. For more information, see for example [LLMSSZ2013].

  INPUT:

  - `t` – a weak \(k\)-tableau in the specified representation:
    - for the ‘core’ representation `t` is a list of lists where each subtableaux should have a \(k+1\)-core shape; `None` is allowed as an entry for skew weak \(k\)-tableaux
    - for the ‘bounded’ representation `t` is a list of lists where each subtableaux should have a \(k\)-bounded shape; `None` is allowed as an entry for skew weak \(k\)-tableaux
    - for the ‘factorized_permutation’ representation `t` is either a list of cyclically decreasing Weyl group elements or a list of reduced words of cyclically decreasing Weyl group elements; to indicate a skew tableau in this representation, `inner_shape` should be the inner shape as a \((k+1)\)-core
  - `k` – positive integer
  - `inner_shape` – this entry is only relevant for the ‘factorized_permutation’ representation and specifies the inner shape in case the tableau is skew (default: `[]`)
  - `representation` – ‘core’, ‘bounded’, or ‘factorized_permutation’ (default: ‘core’)

  EXAMPLES:

  Here is an example of a weak 3-tableau in core representation:

      sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
      sage: t.shape()
      [5, 2, 1]
      sage: t.weight()
      (2, 2, 2)
      sage: type(t)
      <class 'sage.combinat.k_tableau.WeakTableaux_core_with_category.element_class'>

  And now we give a skew weak 3-tableau in core representation:

      sage: ts = WeakTableau([[None, 1, 1, 2, 2], [None, 2], [1]], 3)
      sage: ts.shape()
      ([5, 2, 1], [1, 1])
      sage: ts.weight()
      (2, 2)
      sage: type(ts)
      <class 'sage.combinat.k_tableau.WeakTableaux_core_with_category.element_class'>

  Next we create the analogue of the first example in bounded representation:

      sage: tb = WeakTableau([[1,1,2],[2,3],[3]], 3, representation="bounded")
      sage: tb.shape()
      [3, 2, 1]
      sage: tb.weight()
      (2, 2, 2)
      sage: type(tb)
      <class 'sage.combinat.k_tableau.WeakTableaux_bounded_with_category.element_class'>
      sage: tb.to_core_tableau()
      [[1, 1, 2, 2, 3], [2, 3], [3]]
      sage: t == tb.to_core_tableau()
      True

  And the analogue of the skew example in bounded representation:

      sage: tbs = WeakTableau([[None, 1, 2], [None, 2], [1]], 3, representation = "bounded")
      sage: tbs.shape()
      ([3, 2, 1], [1, 1])
      sage: tbs.weight()
      (2, 2)
      sage: tbs.to_core_tableau()
      [[None, 1, 1, 2, 2], [None, 2], [1]]
      sage: ts.to_bounded_tableau() == tbs
      True

  Finally we do the same examples for the factorized permutation representation:

      sage: tf = WeakTableau([[2,0],[3,2],[1,0]], 3, representation = "factorized_permutation")
      sage: tf.shape()
      [5, 2, 1]
      sage: tf.weight()
      (2, 2, 2)
      sage: type(tf)
      <class 'sage.combinat.k_tableau.WeakTableaux_factorized_permutation_with_category.element_class'>
      sage: tf.to_core_tableau() == t
      True

      sage: tfs = WeakTableau([[0,3],[2,1]], 3, inner_shape = [1,1], representation = 'factorized_permutation')
      sage: tfs.shape()
      ([5, 2, 1], [1, 1])
      sage: tfs.weight()
      (2, 2)
      sage: type(tfs)
      <class 'sage.combinat.k_tableau.WeakTableaux_factorized_permutation_with_category.element_class'>
      sage: tfs.to_core_tableau()
      [[None, 1, 1, 2, 2], [None, 2], [1]]

  Another way to pass from one representation to another is as follows:

      sage: ts
      [[None, 1, 1, 2, 2], [None, 2], [1]]
      sage: ts.parent()._representation
      'core'
      sage: ts.representation('bounded')
      [[None, 1, 2], [None, 2], [1]]

  To test whether a given semistandard tableau is a weak \(k\)-tableau in the bounded representation, one can ask:

      sage: t = Tableau([[1,1,2],[2,3],[3]])
      sage: t.is_k_tableau(3)
      True
      sage: t = SkewTableau([[None, 1, 2], [None, 2], [1]])
      sage: t.is_k_tableau(3)
      True
      sage: t = SkewTableau([[None, 1, 1], [None, 2], [2]])
      sage: t.is_k_tableau(3)
      False

  TESTS:

      sage: t = WeakTableau([[2,0],[3,2],[1,0]], 3, representation = "bla")
      Traceback (most recent call last):
      ...
      NotImplementedError: The representation option needs to be 'core', 'bounded', or 'factorized_permuation'

class `sage.combinat.k_tableau.``WeakTableau_abstract`

  Bases: `sage.structure.list_clone.ClonableList`

  Abstract class for the various element classes of WeakTableau.

  `intermediate_shapes`()

    Return the intermediate shapes of `self`.

    A (skew) tableau with letters \(1,2,\ldots,\ell\) can be viewed as a sequence of shapes, where the \(i\)-th shape is given by the shape of the subtableau on letters \(1,2,\ldots,i\). The output is the list of these shapes.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        sage: t.intermediate_shapes()
        [[], [2], [4, 1], [5, 2, 1]]

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.intermediate_shapes()
        [[2], [2, 1], [3, 1, 1], [4, 1, 1], [5, 2, 1]]

        sage: t = WeakTableau([[1,1,1],[2,2],[3]], 3, representation = 'bounded')
        sage: t.intermediate_shapes()
        [[], [3], [3, 2], [3, 2, 1]]

        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.intermediate_shapes()
        [[2], [3], [3, 1], [3, 1, 1], [3, 2, 1]]

        sage: t = WeakTableau([[0],[3],[2],[3]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.intermediate_shapes()
        [[2], [2, 1], [3, 1, 1], [4, 1, 1], [5, 2, 1]]

  `pp`()

    Return a pretty print string of the tableau.

    EXAMPLES:

        sage: t = WeakTableau([[None, 1, 1, 2, 2], [None, 2], [1]], 3)
        sage: t.pp()
        .  1  1  2  2
        .  2
        1
        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.pp()
        [s2*s0, s3*s2]

  `representation`(representation='core')

    Return the analogue of `self` in the specified representation.

    INPUT:

    - `representation` – ‘core’, ‘bounded’, or ‘factorized_permutation’ (default: ‘core’)

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        sage: t.parent()._representation
        'core'
        sage: t.representation('bounded')
        [[1, 1, 2, 4], [2, 3, 5], [3, 4], [5, 6], [6], [7]]
        sage: t.representation('factorized_permutation')
        [s0, s3*s1, s2*s1, s0*s4, s3*s0, s4*s2, s1*s0]

        sage: tb = WeakTableau([[1, 1, 2, 4], [2, 3, 5], [3, 4], [5, 6], [6], [7]], 4, representation = 'bounded')
        sage: tb.parent()._representation
        'bounded'
        sage: tb.representation('core') == t
        True
        sage: tb.representation('factorized_permutation')
        [s0, s3*s1, s2*s1, s0*s4, s3*s0, s4*s2, s1*s0]

        sage: tp = WeakTableau([[0],[3,1],[2,1],[0,4],[3,0],[4,2],[1,0]], 4, representation = 'factorized_permutation')
        sage: tp.parent()._representation
        'factorized_permutation'
        sage: tp.representation('core') == t
        True
        sage: tp.representation('bounded') == tb
        True

  `shape`()

    Return the shape of `self`.

    When the tableau is straight, the outer shape is returned. When the tableau is skew, the tuple of the outer and inner shape is returned.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.shape()
        [5, 2, 1]
        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.shape()
        ([5, 2, 1], [2])

        sage: t = WeakTableau([[1,1,1],[2,2],[3]], 3, representation = 'bounded')
        sage: t.shape()
        [3, 2, 1]
        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.shape()
        ([3, 2, 1], [2])

        sage: t = WeakTableau([[2],[0,3],[2,1,0]], 3, representation = 'factorized_permutation')
        sage: t.shape()
        [5, 2, 1]
        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.shape()
        ([5, 2, 1], [2])

  `size`()

    Return the size of the shape of `self`.

    In the bounded representation, the size of the shape is the number of boxes in the outer shape minus the number of boxes in the inner shape. For the core and factorized permutation representation, the size is the length of the outer shape minus the length of the inner shape.

    See also

    `sage.combinat.core.Core.length()`

    EXAMPLES:

        sage: t = WeakTableau([[None, 1, 1, 2, 2], [None, 2], [1]], 3)
        sage: t.shape()
        ([5, 2, 1], [1, 1])
        sage: t.size()
        4
        sage: t = WeakTableau([[1,1,2],[2,3],[3]], 3, representation="bounded")
        sage: t.shape()
        [3, 2, 1]
        sage: t.size()
        6

  `weight`()

    Return the weight of `self`.

    The weight is a tuple whose \(i\)-th entry is the number of labels \(i\) in the bounded representation of `self`.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.weight()
        (2, 2, 2)
        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.weight()
        (1, 1, 1, 1)
        sage: t = WeakTableau([[None,2,3],[3]],2)
        sage: t.weight()
        (0, 1, 1)

        sage: t = WeakTableau([[1,1,1],[2,2],[3]], 3, representation = 'bounded')
        sage: t.weight()
        (3, 2, 1)
        sage: t = WeakTableau([[1,1,2],[2,3],[3]], 3, representation = 'bounded')
        sage: t.weight()
        (2, 2, 2)
        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.weight()
        (1, 1, 1, 1)

        sage: t = WeakTableau([[2],[0,3],[2,1,0]], 3, representation = 'factorized_permutation')
        sage: t.weight()
        (3, 2, 1)
        sage: t = WeakTableau([[2,0],[3,2],[1,0]], 3, representation = 'factorized_permutation')
        sage: t.weight()
        (2, 2, 2)
        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.weight()
        (2, 2)

class `sage.combinat.k_tableau.``WeakTableau_bounded`(parent, t)

  Bases: `sage.combinat.k_tableau.WeakTableau_abstract`

  A (skew) weak \(k\)-tableau represented in terms of \(k\)-bounded partitions.

  `check`()

    Check that `self` is a valid weak \(k\)-tableau.

    EXAMPLES:

        sage: t = WeakTableau([[1,1],[2]], 2, representation = 'bounded')
        sage: t.check()

        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.check()

    TESTS:

        sage: t = WeakTableau([[1,1,3],[2,2],[3]], 3, representation = 'bounded')
        Traceback (most recent call last):
        ...
        ValueError: This is not a proper weak 3-tableau

        sage: T = WeakTableaux(3, [3,1], [2,1], representation = 'bounded')
        sage: t = T([[None, 1,1], [2]])
        Traceback (most recent call last):
        ...
        ValueError: The inner shape of the parent does not agree with the inner shape of the tableau!

        sage: t = WeakTableau([[1,1],[1]], 3, representation = 'bounded')
        Traceback (most recent call last):
        ...
        ValueError: The tableaux is not semistandard!

  classmethod `from_core_tableau`(t, k)

    Construct weak \(k\)-bounded tableau from in \(k\)-core tableau.

    EXAMPLES:

        sage: from sage.combinat.k_tableau import WeakTableau_bounded
        sage: WeakTableau_bounded.from_core_tableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        [[1, 1, 2], [2, 3], [3]]

        sage: WeakTableau_bounded.from_core_tableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        [[None, None, 3], [1, 4], [2]]

        sage: WeakTableau_bounded.from_core_tableau([[None,2,3],[3]], 2)
        [[None, 2], [3]]

  `k_charge`(algorithm='I')

    Return the \(k\)-charge of `self`.

    INPUT:

    - `algorithm` – (default: “I”) if “I”, computes \(k\)-charge using the \(I\) algorithm, otherwise uses the \(J\)-algorithm

    OUTPUT:

    - a nonnegative integer

    For the definition of \(k\)-charge and the various algorithms to compute it see Section 3.3 of [LLMSSZ2013].

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2], [2, 3], [3]], 3, representation = 'bounded')
        sage: t.k_charge()
        2
        sage: t = WeakTableau([[1, 3, 5], [2, 6], [4]], 3, representation = 'bounded')
        sage: t.k_charge()
        8
        sage: t = WeakTableau([[1, 1, 2, 4], [2, 3, 5], [3, 4], [5, 6], [6], [7]], 4, representation = 'bounded')
        sage: t.k_charge()
        12

  `shape_bounded`()

    Return the shape of `self` as \(k\)-bounded partition.

    When the tableau is straight, the outer shape is returned as a \(k\)-bounded partition. When the tableau is skew, the tuple of the outer and inner shape is returned as \(k\)-bounded partitions.

    EXAMPLES:

        sage: t = WeakTableau([[1,1,1],[2,2],[3]], 3, representation = 'bounded')
        sage: t.shape_bounded()
        [3, 2, 1]

        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.shape_bounded()
        ([3, 2, 1], [2])

  `shape_core`()

    Return the shape of `self` as \((k+1)\)-core.

    When the tableau is straight, the outer shape is returned as a \((k+1)\)-core. When the tableau is skew, the tuple of the outer and inner shape is returned as \((k+1)\)-cores.

    EXAMPLES:

        sage: t = WeakTableau([[1,1,1],[2,2],[3]], 3, representation = 'bounded')
        sage: t.shape_core()
        [5, 2, 1]

        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.shape_core()
        ([5, 2, 1], [2])

  `to_core_tableau`()

    Return the weak \(k\)-tableau `self` where the shape of each restricted tableau is a \((k+1)\)-core.

    EXAMPLES:

        sage: t = WeakTableau([[1,1,2,4],[2,3,5],[3,4],[5,6],[6],[7]], 4, representation = 'bounded')
        sage: c = t.to_core_tableau(); c
        [[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]]
        sage: type(c)
        <class 'sage.combinat.k_tableau.WeakTableaux_core_with_category.element_class'>
        sage: t = WeakTableau([], 4, representation = 'bounded')
        sage: t.to_core_tableau()
        []

        sage: from sage.combinat.k_tableau import WeakTableau_bounded
        sage: t = WeakTableau([[1,1,2],[2,3],[3]], 3, representation = 'bounded')
        sage: WeakTableau_bounded.from_core_tableau(t.to_core_tableau(),3)
        [[1, 1, 2], [2, 3], [3]]
        sage: t == WeakTableau_bounded.from_core_tableau(t.to_core_tableau(),3)
        True

        sage: t = WeakTableau([[None, None, 1], [2, 4], [3]], 3, representation = 'bounded')
        sage: t.to_core_tableau()
        [[None, None, 1, 2, 4], [2, 4], [3]]
        sage: t == WeakTableau_bounded.from_core_tableau(t.to_core_tableau(),3)
        True

class `sage.combinat.k_tableau.``WeakTableau_core`(parent, t)

  Bases: `sage.combinat.k_tableau.WeakTableau_abstract`

  A (skew) weak \(k\)-tableau represented in terms of \((k+1)\)-cores.

  `check`()

    Check that `self` is a valid weak \(k\)-tableau.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2], [2]], 2)
        sage: t.check()
        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.check()

    TESTS:

        sage: T = WeakTableaux(2, [3,1], [1,1,1,1])
        sage: t = T([[1,2,3],[3]])
        Traceback (most recent call last):
        ...
        ValueError: The weight of the parent does not agree with the weight of the tableau!

        sage: t = WeakTableau([[1, 2, 2], [1]], 2)
        Traceback (most recent call last):
        ...
        ValueError: The tableau is not semistandard!

  `dictionary_of_coordinates_at_residues`(v)

    Return a dictionary assigning to all residues of `self` with label `v` a list of cells with the given residue.

    INPUT:

    - `v` – a label of a cell in `self`

    OUTPUT:

    - dictionary assigning coordinates in `self` to residues

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        sage: t.dictionary_of_coordinates_at_residues(3)
        {0: [(0, 4), (1, 1)], 2: [(2, 0)]}

        sage: t = WeakTableau([[None, None, 1, 1, 4], [1, 4], [3]], 3)
        sage: t.dictionary_of_coordinates_at_residues(1)
        {2: [(0, 2)], 3: [(0, 3), (1, 0)]}

        sage: t = WeakTableau([], 3)
        sage: t.dictionary_of_coordinates_at_residues(1)
        {}

  `k_charge`(algorithm='I')

    Return the \(k\)-charge of `self`.

    INPUT:

    - `algorithm` – (default: “I”) if “I”, computes \(k\)-charge using the \(I\) algorithm, otherwise uses the \(J\)-algorithm

    OUTPUT:

    - a nonnegative integer

    For the definition of \(k\)-charge and the various algorithms to compute it see Section 3.3 of [LLMSSZ2013].

    See also

    `k_charge_I()` and `k_charge_J()`

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.k_charge()
        2
        sage: t = WeakTableau([[1, 3, 4, 5, 6], [2, 6], [4]], 3)
        sage: t.k_charge()
        8
        sage: t = WeakTableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        sage: t.k_charge()
        12

    TESTS:

        sage: T = WeakTableaux(4, [13,9,5,3,2,1,1], [4,3,3,2,2,1,1,1])
        sage: T.cardinality()
        6
        sage: all(t.k_charge_I() == t.k_charge_J() for t in T)
        True

  `k_charge_I`()

    Return the \(k\)-charge of `self` using the \(I\)-algorithm.

    For the definition of \(k\)-charge and the \(I\)-algorithm see Section 3.3 of [LLMSSZ2013].

    OUTPUT:

    - a nonnegative integer

    See also

    `k_charge()` and `k_charge_J()`

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.k_charge_I()
        2
        sage: t = WeakTableau([[1, 3, 4, 5, 6], [2, 6], [4]], 3)
        sage: t.k_charge_I()
        8
        sage: t = WeakTableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        sage: t.k_charge_I()
        12

    TESTS:

        sage: t = WeakTableau([[None, None, 1, 1, 4], [1, 4], [3]], 3)
        sage: t.k_charge_I()
        Traceback (most recent call last):
        ...
        ValueError: k-charge is not defined for skew weak tableaux

  `k_charge_J`()

    Return the \(k\)-charge of `self` using the \(J\)-algorithm.

    For the definition of \(k\)-charge and the \(J\)-algorithm see Section 3.3 of [LLMSSZ2013].

    OUTPUT:

    - a nonnegative integer

    See also

    `k_charge()` and `k_charge_I()`

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.k_charge_J()
        2
        sage: t = WeakTableau([[1, 3, 4, 5, 6], [2, 6], [4]], 3)
        sage: t.k_charge_J()
        8
        sage: t = WeakTableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        sage: t.k_charge_J()
        12

    TESTS:

        sage: t = WeakTableau([[None, None, 1, 1, 4], [1, 4], [3]], 3)
        sage: t.k_charge_I()
        Traceback (most recent call last):
        ...
        ValueError: k-charge is not defined for skew weak tableaux

        sage: t = WeakTableau([[1, 1, 2, 3], [2, 4, 4], [3, 6], [5]], 4, representation='bounded')
        sage: t.k_charge() == t.k_charge(algorithm = 'J')
        True

  `list_of_standard_cells`()

    Return a list of lists of the coordinates of the standard cells of `self`.

    INPUT:

    - `self` – a weak \(k\)-tableau in core representation with partition weight

    OUTPUT:

    - a list of lists of coordinates

    Warning

    This method currently only works for straight weak tableaux with partition weight.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: t.list_of_standard_cells()
        [[(0, 1), (1, 0), (2, 0)], [(0, 0), (0, 2), (1, 1)]]
        sage: t = WeakTableau([[1, 1, 1, 2], [2, 2, 3]], 5)
        sage: t.list_of_standard_cells()
        [[(0, 2), (1, 1), (1, 2)], [(0, 1), (1, 0)], [(0, 0), (0, 3)]]
        sage: t = WeakTableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        sage: t.list_of_standard_cells()
        [[(0, 1), (1, 0), (2, 0), (0, 5), (3, 0), (4, 0), (5, 0)], [(0, 0), (0, 2), (1, 1), (2, 1), (1, 2), (3, 1)]]

    TESTS:

        sage: t = WeakTableau([],3)
        sage: t.list_of_standard_cells()
        []

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.list_of_standard_cells()
        Traceback (most recent call last):
        ...
        ValueError: This method only works for straight tableaux!

        sage: t = WeakTableau([[1,2],[2]], 3)
        sage: t.list_of_standard_cells()
        Traceback (most recent call last):
        ...
        ValueError: This method only works for weak tableaux with partition weight!

  `residues_of_entries`(v)

    Return a list of residues of cells of weak \(k\)-tableau `self` labeled by `v`.

    INPUT:

    - `v` – a label of a cell in `self`

    OUTPUT:

    - a list of residues

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        sage: t.residues_of_entries(1)
        [0, 1]

        sage: t = WeakTableau([[None, None, 1, 1, 4], [1, 4], [3]], 3)
        sage: t.residues_of_entries(1)
        [2, 3]

  `shape_bounded`()

    Return the shape of `self` as a \(k\)-bounded partition.

    When the tableau is straight, the outer shape is returned as a \(k\)-bounded partition. When the tableau is skew, the tuple of the outer and inner shape is returned as \(k\)-bounded partitions.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        sage: t.shape_bounded()
        [3, 2, 1]

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.shape_bounded()
        ([3, 2, 1], [2])

  `shape_core`()

    Return the shape of `self` as a \((k+1)\)-core.

    When the tableau is straight, the outer shape is returned as a core. When the tableau is skew, the tuple of the outer and inner shape is returned as cores.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        sage: t.shape_core()
        [5, 2, 1]

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.shape_core()
        ([5, 2, 1], [2])

  `to_bounded_tableau`()

    Return the bounded representation of the weak \(k\)-tableau `self`.

    Each restricted sutableaux of the output is a \(k\)-bounded partition.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: c = t.to_bounded_tableau(); c
        [[1, 1, 2], [2, 3], [3]]
        sage: type(c)
        <class 'sage.combinat.k_tableau.WeakTableaux_bounded_with_category.element_class'>

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: t.to_bounded_tableau()
        [[None, None, 3], [1, 4], [2]]
        sage: t.to_bounded_tableau().to_core_tableau() == t
        True

  `to_factorized_permutation_tableau`()

    Return the factorized permutation representation of the weak \(k\)-tableau `self`.

    EXAMPLES:

        sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]], 3)
        sage: c = t.to_factorized_permutation_tableau(); c
        [s2*s0, s3*s2, s1*s0]
        sage: type(c)
        <class 'sage.combinat.k_tableau.WeakTableaux_factorized_permutation_with_category.element_class'>
        sage: c.to_core_tableau() == t
        True

        sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
        sage: c = t.to_factorized_permutation_tableau(); c
        [s0, s3, s2, s3]
        sage: c._inner_shape
        [2]
        sage: c.to_core_tableau() == t
        True

    TESTS:

        sage: t = WeakTableau([], 4)
        sage: c = t.to_factorized_permutation_tableau(); c
        [1]
        sage: c._inner_shape
        []
        sage: c.to_core_tableau() == t
        True

class `sage.combinat.k_tableau.``WeakTableau_factorized_permutation`(parent, t)

  Bases: `sage.combinat.k_tableau.WeakTableau_abstract`

  A weak (skew) \(k\)-tableau represented in terms of factorizations of affine permutations into cyclically decreasing elements.

  `check`()

    Check that `self` is a valid weak \(k\)-tableau.

    EXAMPLES:

        sage: t = WeakTableau([[2],[0,3],[2,1,0]], 3, representation = 'factorized_permutation')
        sage: t.check()

    TESTS:

        sage: t = WeakTableau([[2,0],[3,2]], 3, representation = 'factorized_permutation')
        Traceback (most recent call last):
        ...
        ValueError: Error! this only works on type 'A' affine Grassmannian elements

        sage: T = WeakTableaux(3, [4,1], [2,1], representation = 'factorized_permutation')
        sage: t = T([[2],[1],[0]])
        Traceback (most recent call last):
        ...
        ValueError: The weight of the parent does not agree with the weight of the tableau!

  classmethod `from_core_tableau`(t, k)

    Construct weak factorized affine permutation tableau from a \(k\)-core tableau.

    EXAMPLES:

        sage: from sage.combinat.k_tableau import WeakTableau_factorized_permutation
        sage: WeakTableau_factorized_permutation.from_core_tableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
        [s2*s0, s3*s2, s1*s0]
        sage: WeakTableau_factorized_permutation.from_core_tableau([[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]], 4)
        [s0, s3*s1, s2*s1, s0*s4, s3*s0, s4*s2, s1*s0]
        sage: WeakTableau_factorized_permutation.from_core_tableau([[None, 1, 1, 2, 2], [None, 2], [1]], 3)
        [s0*s3, s2*s1]

  `k_charge`(algorithm='I')

    Return the \(k\)-charge of `self`.

    OUTPUT:

    - a nonnegative integer

    EXAMPLES:

        sage: t = WeakTableau([[2,0],[3,2],[1,0]], 3, representation = 'factorized_permutation')
        sage: t.k_charge()
        2
        sage: t = WeakTableau([[0],[3],[2],[1],[3],[0]], 3, representation = 'factorized_permutation')
        sage: t.k_charge()
        8
        sage: t = WeakTableau([[0],[3,1],[2,1],[0,4],[3,0],[4,2],[1,0]], 4, representation = 'factorized_permutation')
        sage: t.k_charge()
        12

  `shape_bounded`()

    Return the shape of `self` as a \(k\)-bounded partition.

    When the tableau is straight, the outer shape is returned as a \(k\)-bounded partition. When the tableau is skew, the tuple of the outer and inner shape is returned as \(k\)-bounded partitions.

    EXAMPLES:

        sage: t = WeakTableau([[2],[0,3],[2,1,0]], 3, representation = 'factorized_permutation')
        sage: t.shape_bounded()
        [3, 2, 1]

        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.shape_bounded()
        ([3, 2, 1], [2])

  `shape_core`()

    Return the shape of `self` as a \((k+1)\)-core.

    When the tableau is straight, the outer shape is returned as a core. When the tableau is skew, the tuple of the outer and inner shape is returned as cores.

    EXAMPLES:

        sage: t = WeakTableau([[2],[0,3],[2,1,0]], 3, representation = 'factorized_permutation')
        sage: t.shape_core()
        [5, 2, 1]

        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.shape()
        ([5, 2, 1], [2])

  static `straighten_input`(t, k)

    Straightens input.

    INPUT:

    - `t` – a list of reduced words or a list of elements in the Weyl group of type \(A_k^{(1)}\)
    - `k` – a positive integer

    EXAMPLES:

        sage: from sage.combinat.k_tableau import WeakTableau_factorized_permutation
        sage: WeakTableau_factorized_permutation.straighten_input([[2,0],[3,2],[1,0]], 3)
        (s2*s0, s3*s2, s1*s0)
        sage: W = WeylGroup(['A',4,1])
        sage: WeakTableau_factorized_permutation.straighten_input([W.an_element(),W.an_element()], 4)
        (s0*s1*s2*s3*s4, s0*s1*s2*s3*s4)

    TESTS:

        sage: WeakTableau_factorized_permutation.straighten_input([W.an_element(),W.an_element()], 3)
        Traceback (most recent call last):
        ...
        ValueError: a matrix from Full MatrixSpace of 5 by 5 dense matrices over Rational Field cannot be converted to a matrix in Full MatrixSpace of 4 by 4 dense matrices over Rational Field!

  `to_core_tableau`()

    Return the weak \(k\)-tableau `self` where the shape of each restricted tableau is a \((k+1)\)-core.

    EXAMPLES:

        sage: t = WeakTableau([[0], [3,1], [2,1], [0,4], [3,0], [4,2], [1,0]], 4, representation = 'factorized_permutation'); t
        [s0, s3*s1, s2*s1, s0*s4, s3*s0, s4*s2, s1*s0]
        sage: c = t.to_core_tableau(); c
        [[1, 1, 2, 3, 4, 4, 5, 5, 6], [2, 3, 5, 5, 6], [3, 4, 7], [5, 6], [6], [7]]
        sage: type(c)
        <class 'sage.combinat.k_tableau.WeakTableaux_core_with_category.element_class'>
        sage: t = WeakTableau([[]], 4, representation = 'factorized_permutation'); t
        [1]
        sage: t.to_core_tableau()
        []

        sage: from sage.combinat.k_tableau import WeakTableau_factorized_permutation
        sage: t = WeakTableau([[2,0],[3,2],[1,0]], 3, representation = 'factorized_permutation')
        sage: WeakTableau_factorized_permutation.from_core_tableau(t.to_core_tableau(), 3)
        [s2*s0, s3*s2, s1*s0]
        sage: t == WeakTableau_factorized_permutation.from_core_tableau(t.to_core_tableau(), 3)
        True

        sage: t = WeakTableau([[2,0],[3,2]], 3, inner_shape = [2], representation = 'factorized_permutation')
        sage: t.to_core_tableau()
        [[None, None, 1, 1, 2], [1, 2], [2]]
        sage: t == WeakTableau_factorized_permutation.from_core_tableau(t.to_core_tableau(), 3)
        True

`sage.combinat.k_tableau.``WeakTableaux`(k, shape, weight, representation='core')

  This is the dispatcher method for the parent class of weak \(k\)-tableaux.

  INPUT:

  - `k` – positive integer
  - `shape` – shape of the weak \(k\)-tableaux; for the ‘core’ and ‘factorized_permutation’ representation, the shape is inputted as a \((k+1)\)-core; for the ‘bounded’ representation, the shape is inputted as a \(k\)-bounded partition; for skew tableaux, the shape is inputted as a tuple of the outer and inner shape
  - `weight` – the weight of the weak \(k\)-tableaux as a list or tuple
  - `representation` – ‘core’, ‘bounded’, or ‘factorized_permutation’ (default: ‘core’)

  EXAMPLES:

      sage: T = WeakTableaux(3, [5,2,1], [1,1,1,1,1,1])
      sage: T.list()
      [[[1, 3, 4, 5, 6], [2, 6], [4]],
      [[1, 2, 4, 5, 6], [3, 6], [4]],
      [[1, 2, 3, 4, 6], [4, 6], [5]],
      [[1, 2, 3, 4, 5], [4, 5], [6]]]
      sage: T.cardinality()
      4

      sage: T = WeakTableaux(3, [[5,2,1], [2]], [1,1,1,1])
      sage: T.list()
      [[[None, None, 2, 3, 4], [1, 4], [2]],
      [[None, None, 1, 2, 4], [2, 4], [3]],
      [[None, None, 1, 2, 3], [2, 3], [4]]]

      sage: T = WeakTableaux(3, [3,2,1], [1,1,1,1,1,1], representation = 'bounded')
      sage: T.list()
      [[[1, 3, 5], [2, 6], [4]],
      [[1, 2, 5], [3, 6], [4]],
      [[1, 2, 3], [4, 6], [5]],
      [[1, 2, 3], [4, 5], [6]]]

      sage: T = WeakTableaux(3, [[3,2,1], [2]], [1,1,1,1], representation = 'bounded')
      sage: T.list()
      [[[None, None, 3], [1, 4], [2]],
      [[None, None, 1], [2, 4], [3]],
      [[None, None, 1], [2, 3], [4]]]

      sage: T = WeakTableaux(3, [5,2,1], [1,1,1,1,1,1], representation = 'factorized_permutation')
      sage: T.list()
      [[s0, s3, s2, s1, s3, s0],
      [s0, s3, s2, s3, s1, s0],
      [s0, s2, s3, s2, s1, s0],
      [s2, s0, s3, s2, s1, s0]]

      sage: T = WeakTableaux(3, [[5,2,1], [2]], [1,1,1,1], representation = 'factorized_permutation')
      sage: T.list()
      [[s0, s3, s2, s3], [s0, s2, s3, s2], [s2, s0, s3, s2]]

class `sage.combinat.k_tableau.``WeakTableaux_abstract`

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

  Abstract class for the various parent classes of WeakTableaux.

  `representation`(representation='core')

    Return the analogue of `self` in the specified representation.

    INPUT:

    - `representation` – ‘core’, ‘bounded’, or ‘factorized_permutation’ (default: ‘core’)

    EXAMPLES:

        sage: T = WeakTableaux(3, [5,2,1], [1,1,1,1,1,1])
        sage: T._representation
        'core'
        sage: T.representation('bounded')
        Bounded weak 3-Tableaux of (skew) 3-bounded shape [3, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('factorized_permutation')
        Factorized permutation (skew) weak 3-Tableaux of shape [5, 2, 1] and weight (1, 1, 1, 1, 1, 1)

        sage: T = WeakTableaux(3, [3,2,1], [1,1,1,1,1,1], representation = 'bounded')
        sage: T._representation
        'bounded'
        sage: T.representation('core')
        Core weak 3-Tableaux of (skew) core shape [5, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('bounded')
        Bounded weak 3-Tableaux of (skew) 3-bounded shape [3, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('bounded') == T
        True
        sage: T.representation('factorized_permutation')
        Factorized permutation (skew) weak 3-Tableaux of shape [5, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('factorized_permutation') == T
        False

        sage: T = WeakTableaux(3, [5,2,1], [1,1,1,1,1,1], representation = 'factorized_permutation')
        sage: T._representation
        'factorized_permutation'
        sage: T.representation('core')
        Core weak 3-Tableaux of (skew) core shape [5, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('bounded')
        Bounded weak 3-Tableaux of (skew) 3-bounded shape [3, 2, 1] and weight (1, 1, 1, 1, 1, 1)
        sage: T.representation('factorized_permutation')
        Factorized permutation (skew) weak 3-Tableaux of shape [5, 2, 1] and weight (1, 1, 1, 1, 1, 1)

  `shape`()

    Return the shape of the tableaux of `self`.

    When `self` is the class of straight tableaux, the outer shape is returned. When `self` is the class of skew tableaux, the tuple of the outer and inner shape is returned.

    Note that in the ‘core’ and ‘factorized_permutation’ representation, the shapes are \((k+1)\)-cores. In the ‘bounded’ representation, the shapes are \(k\)-bounded partitions.

    If the user wants to access the skew shape (even if the inner shape is empty), please use `self._shape`.

    EXAMPLES:

        sage: T = WeakTableaux(3, [5,2,2], [2,2,2,1])
        sage: T.shape()
        [5, 2, 2]
        sage: T._shape
        ([5, 2, 2], [])
        sage: T = WeakTableaux(3, [[5,2,2], [1]], [2,1,2,1])
        sage: T.shape()
        ([5, 2, 2], [1])

        sage: T = WeakTableaux(3, [3,2,2], [2,2,2,1], representation = 'bounded')
        sage: T.shape()
        [3, 2, 2]
        sage: T._shape
        ([3, 2, 2], [])
        sage: T = WeakTableaux(3, [[3,2,2], [1]], [2,1,2,1], representation = 'bounded')
        sage: T.shape()
        ([3, 2, 2], [1])

        sage: T = WeakTableaux(3, [4,1], [2,2], representation = 'factorized_permutation')
        sage: T.shape()
        [4, 1]
        sage: T._shape
        ([4, 1], [])
        sage: T = WeakTableaux(4, [[6,2,1], [2]], [2,1,1,1], representation = 'factorized_permutation')
        sage: T.shape()
        ([6, 2, 1], [2])

  `size`()

    Return the size of the shape.

    In the bounded representation, the size of the shape is the number of boxes in the outer shape minus the number of boxes in the inner shape. For the core and factorized permutation representation, the size is the length of the outer shape minus the length of the inner shape.

    EXAMPLES:

        sage: T = WeakTableaux(3, [5,2,1], [1,1,1,1,1,1])
        sage: T.size()
        6
        sage: T = WeakTableaux(3, [3,2,1], [1,1,1,1,1,1], representation = 'bounded')
        sage: T.size()
        6
        sage: T = WeakTableaux(4, [[6,2,1], [2]], [2,1,1,1], 'factorized_permutation')
        sage: T.size()
        5

class `sage.combinat.k_tableau.``WeakTableaux_bounded`(k, shape, weight)

  Bases: `sage.combinat.k_tableau.WeakTableaux_abstract`

  The class of (skew) weak \(k\)-tableaux in the bounded representation of shape `shape` (as \(k\)-bounded partition or tuple of \(k\)-bounded partitions in the skew case) and weight `weight`.

  INPUT:

  - `k` – positive integer
  - `shape` – the shape of the \(k\)-tableaux represented as a \(k\)-bounded partition; if the tableaux are skew, the shape is a tuple of the outer and inner shape each represented as a \(k\)-bounded partition
  - `weight` – the weight of the \(k\)-tableaux

  EXAMPLES:

      sage: T = WeakTableaux(3, [3,1], [2,2], representation = 'bounded')
      sage: T.list()
      [[[1, 1, 2], [2]]]

      sage: T = WeakTableaux(3, [[3,2,1], [2]], [1,1,1,1], representation = 'bounded')
      sage: T.list()
      [[[None, None, 3], [1, 4], [2]],
      [[None, None, 1], [2, 4], [3]],
      [[None, None, 1], [2, 3], [4]]]

  `Element`

    alias of `WeakTableau_bounded`

class `sage.combinat.k_tableau.``WeakTableaux_core`(k, shape, weight)

  Bases: `sage.combinat.k_tableau.WeakTableaux_abstract`

  The class of (skew) weak \(k\)-tableaux in the core representation of shape `shape` (as \(k+1\)-core) and weight `weight`.

  INPUT:

  - `k` – positive integer
  - `shape` – the shape of the \(k\)-tableaux represented as a \((k+1)\)-core; if the tableaux are skew, the shape is a tuple of the outer and inner shape (both as \((k+1)\)-cores)
  - `weight` – the weight of the \(k\)-tableaux

  EXAMPLES:

      sage: T = WeakTableaux(3, [4,1], [2,2])
      sage: T.list()
      [[[1, 1, 2, 2], [2]]]

      sage: T = WeakTableaux(3, [[5,2,1], [2]], [1,1,1,1])
      sage: T.list()
      [[[None, None, 2, 3, 4], [1, 4], [2]],
      [[None, None, 1, 2, 4], [2, 4], [3]],
      [[None, None, 1, 2, 3], [2, 3], [4]]]

  `Element`

    alias of `WeakTableau_core`

  `circular_distance`(cr, r)

    Return the shortest counterclockwise distance between `cr` and `r` modulo \(k+1\).

    INPUT:

    - `cr`, `r` – nonnegative integers between \(0\) and \(k\)

    OUTPUT:

    - a positive integer

    EXAMPLES:

        sage: T = WeakTableaux(10, [], [])
        sage: T.circular_distance(8, 6)
        2
        sage: T.circular_distance(8, 8)
        0
        sage: T.circular_distance(8, 9)
        10

  `diag`(c, ha)

    Return the number of diagonals strictly between cells `c` and `ha` of the same residue as `c`.

    INPUT:

    - `c` – a cell in the lattice
    - `ha` – another cell in the lattice with bigger row and smaller column than \(c\)

    OUTPUT:

    - a nonnegative integer

    EXAMPLES:

        sage: T = WeakTableaux(4, [5,2,2], [2,2,2,1])
        sage: T.diag((1,2),(4,0))
        0

class `sage.combinat.k_tableau.``WeakTableaux_factorized_permutation`(k, shape, weight)

  Bases: `sage.combinat.k_tableau.WeakTableaux_abstract`

  The class of (skew) weak \(k\)-tableaux in the factorized permutation representation of shape `shape` (as \(k+1\)-core or tuple of \((k+1)\)-cores in the skew case) and weight `weight`.

  INPUT:

  - `k` – positive integer
  - `shape` – the shape of the \(k\)-tableaux represented as a \((k+1)\)-core; in the skew case the shape is a tuple of the outer and inner shape both as \((k+1)\)-cores
  - `weight` – the weight of the \(k\)-tableaux

  EXAMPLES:

      sage: T = WeakTableaux(3, [4,1], [2,2], representation = 'factorized_permutation')
      sage: T.list()
      [[s3*s2, s1*s0]]

      sage: T = WeakTableaux(4, [[6,2,1], [2]], [2,1,1,1], representation = 'factorized_permutation')
      sage: T.list()
      [[s0, s4, s3, s4*s2], [s0, s3, s4, s3*s2], [s3, s0, s4, s3*s2]]

  `Element`

    alias of `WeakTableau_factorized_permutation`

`sage.combinat.k_tableau.``intermediate_shapes`(t)

  Return the intermediate shapes of tableau `t`.

  A (skew) tableau with letters \(1, 2,\ldots, \ell\) can be viewed as a sequence of shapes, where the \(i\)-th shape is given by the shape of the subtableau on letters \(1, 2, \ldots, i\). The output is the list of these shapes.

  OUTPUT:

  - a list of lists representing partitions

  EXAMPLES:

      sage: from sage.combinat.k_tableau import intermediate_shapes
      sage: t = WeakTableau([[1, 1, 2, 2, 3], [2, 3], [3]],3)
      sage: intermediate_shapes(t)
      [[], [2], [4, 1], [5, 2, 1]]

      sage: t = WeakTableau([[None, None, 2, 3, 4], [1, 4], [2]], 3)
      sage: intermediate_shapes(t)
      [[2], [2, 1], [3, 1, 1], [4, 1, 1], [5, 2, 1]]

`sage.combinat.k_tableau.``nabs`(v)

  Return the absolute value of `v` or `None`.

  INPUT:

  - `v` – either an integer or `None`

  OUTPUT:

  - either a non-negative integer or `None`

  EXAMPLES:

      sage: from sage.combinat.k_tableau import nabs
      sage: nabs(None)
      sage: nabs(-3)
      3
      sage: nabs(None)

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