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# Skew Partitions

A skew partition `skp` of size \(n\) is a pair of partitions \([p_1, p_2]\) where \(p_1\) is a partition of the integer \(n_1\), \(p_2\) is a partition of the integer \(n_2\), \(p_2\) is an inner partition of \(p_1\), and \(n = n_1 - n_2\). We say that \(p_1\) and \(p_2\) are respectively the inner and outer partitions of `skp`.

A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition \(p_1\) which are not in the inner partition \(p_2\) appear in the picture. For example, this is the diagram of the skew partition [[5,4,3,1],[3,3,1]].

    sage: print SkewPartition([[5,4,3,1],[3,3,1]]).diagram()
       **
       *
     **
    *

A skew partition can be connected, which can easily be described in graphic terms: for each pair of consecutive rows, there are at least two cells (one in each row) which have a common edge. This is the diagram of the connected skew partition `[[5,4,3,1],[3,1]]`:

    sage: print SkewPartition([[5,4,3,1],[3,1]]).diagram()
       **
     ***
    ***
    *
    sage: SkewPartition([[5,4,3,1],[3,1]]).is_connected()
    True

The first example of a skew partition is not a connected one.

Applying a reflection with respect to the main diagonal yields the diagram of the conjugate skew partition, here `[[4,3,3,2,1],[3,3,2]]`:

    sage: SkewPartition([[5,4,3,1],[3,3,1]]).conjugate()
    [4, 3, 3, 2, 1] / [3, 2, 2]
    sage: print SkewPartition([[5,4,3,1],[3,3,1]]).conjugate().diagram()
       *
      *
      *
    **
    *

The outer corners of a skew partition are the corners of its outer partition. The inner corners are the internal corners of the outer partition when the inner partition is taken off. Shown below are the coordinates of the inner and outer corners.

    sage: SkewPartition([[5,4,3,1],[3,3,1]]).outer_corners()
    [(0, 4), (1, 3), (2, 2), (3, 0)]
    sage: SkewPartition([[5,4,3,1],[3,3,1]]).inner_corners()
    [(0, 3), (2, 1), (3, 0)]

EXAMPLES:

There are 9 skew partitions of size 3, with no empty row nor empty column:

    sage: SkewPartitions(3).cardinality()
    9
    sage: SkewPartitions(3).list()
    [[3] / [],
     [2, 1] / [],
     [3, 1] / [1],
     [2, 2] / [1],
     [3, 2] / [2],
     [1, 1, 1] / [],
     [2, 2, 1] / [1, 1],
     [2, 1, 1] / [1],
     [3, 2, 1] / [2, 1]]

There are 4 connected skew partitions of size 3:

    sage: SkewPartitions(3, overlap=1).cardinality()
    4
    sage: SkewPartitions(3, overlap=1).list()
    [[3] / [], [2, 1] / [], [2, 2] / [1], [1, 1, 1] / []]

This is the conjugate of the skew partition `[[4,3,1], [2]]`

    sage: SkewPartition([[4,3,1], [2]]).conjugate()
    [3, 2, 2, 1] / [1, 1]

Geometrically, we just applied a reflection with respect to the main diagonal on the diagram of the partition. Of course, this operation is an involution:

    sage: SkewPartition([[4,3,1],[2]]).conjugate().conjugate()
    [4, 3, 1] / [2]

The `jacobi_trudi()` method computes the Jacobi-Trudi matrix. See [Mac95] for a definition and discussion.

    sage: SkewPartition([[4,3,1],[2]]).jacobi_trudi()
    [h[2]  h[]    0]
    [h[5] h[3]  h[]]
    [h[6] h[4] h[1]]

This example shows how to compute the corners of a skew partition.

    sage: SkewPartition([[4,3,1],[2]]).inner_corners()
    [(0, 2), (1, 0)]
    sage: SkewPartition([[4,3,1],[2]]).outer_corners()
    [(0, 3), (1, 2), (2, 0)]

REFERENCES:

[Mac95] | Macdonald I.-G., (1995), “Symmetric Functions and Hall Polynomials”, Oxford Science Publication

AUTHORS:

- Mike Hansen: Initial version
- Travis Scrimshaw (2013-02-11): Factored out `CombinatorialClass`

class `sage.combinat.skew_partition.``SkewPartition`(parent, skp)

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

  A skew partition.

  A skew partition of shape \(\lambda / \mu\) is the Young diagram from the partition \(\lambda\) and removing the partition \(\mu\) from the upper-left corner in English convention.

  `cell_poset`(orientation='SE')

    Return the Young diagram of `self` as a poset. The optional keyword variable `orientation` determines the order relation of the poset.

    The poset always uses the set of cells of the Young diagram of `self` as its ground set. The order relation of the poset depends on the `orientation` variable (which defaults to `"SE"`). Concretely, `orientation` has to be specified to one of the strings `"NW"`, `"NE"`, `"SW"`, and `"SE"`, standing for “northwest”, “northeast”, “southwest” and “southeast”, respectively. If `orientation` is `"SE"`, then the order relation of the poset is such that a cell \(u\) is greater or equal to a cell \(v\) in the poset if and only if \(u\) lies weakly southeast of \(v\) (this means that \(u\) can be reached from \(v\) by a sequence of south and east steps; the sequence is allowed to consist of south steps only, or of east steps only, or even be empty). Similarly the order relation is defined for the other three orientations. The Young diagram is supposed to be drawn in English notation.

    The elements of the poset are the cells of the Young diagram of `self`, written as tuples of zero-based coordinates (so that \((3, 7)\) stands for the \(8\)-th cell of the \(4\)-th row, etc.).

    EXAMPLES:

        sage: p = SkewPartition([[3,3,1], [2,1]])
        sage: Q = p.cell_poset(); Q
        Finite poset containing 4 elements
        sage: sorted(Q)
        [(0, 2), (1, 1), (1, 2), (2, 0)]
        sage: sorted(Q.maximal_elements())
        [(1, 2), (2, 0)]
        sage: sorted(Q.minimal_elements())
        [(0, 2), (1, 1), (2, 0)]
        sage: sorted(Q.upper_covers((1, 1)))
        [(1, 2)]
        sage: sorted(Q.upper_covers((0, 2)))
        [(1, 2)]

        sage: P = p.cell_poset(orientation="NW"); P
        Finite poset containing 4 elements
        sage: sorted(P)
        [(0, 2), (1, 1), (1, 2), (2, 0)]
        sage: sorted(P.minimal_elements())
        [(1, 2), (2, 0)]
        sage: sorted(P.maximal_elements())
        [(0, 2), (1, 1), (2, 0)]
        sage: sorted(P.upper_covers((1, 2)))
        [(0, 2), (1, 1)]

        sage: R = p.cell_poset(orientation="NE"); R
        Finite poset containing 4 elements
        sage: sorted(R)
        [(0, 2), (1, 1), (1, 2), (2, 0)]
        sage: R.maximal_elements()
        [(0, 2)]
        sage: R.minimal_elements()
        [(2, 0)]
        sage: R.upper_covers((2, 0))
        [(1, 1)]
        sage: sorted([len(R.upper_covers(v)) for v in R])
        [0, 1, 1, 1]

    TESTS:

    We check that the posets are really what they should be for size up to \(6\):

        sage: def check_NW(n):
        ....:     for p in SkewPartitions(n):
        ....:         P = p.cell_poset(orientation="NW")
        ....:         for c in p.cells():
        ....:             for d in p.cells():
        ....:                 if P.le(c, d) != (c[0] >= d[0]
        ....:                                   and c[1] >= d[1]):
        ....:                     return False
        ....:     return True
        sage: all( check_NW(n) for n in range(7) )
        True

        sage: def check_NE(n):
        ....:     for p in SkewPartitions(n):
        ....:         P = p.cell_poset(orientation="NE")
        ....:         for c in p.cells():
        ....:             for d in p.cells():
        ....:                 if P.le(c, d) != (c[0] >= d[0]
        ....:                                   and c[1] <= d[1]):
        ....:                     return False
        ....:     return True
        sage: all( check_NE(n) for n in range(7) )
        True

        sage: def test_duality(n, ori1, ori2):
        ....:     for p in SkewPartitions(n):
        ....:         P = p.cell_poset(orientation=ori1)
        ....:         Q = p.cell_poset(orientation=ori2)
        ....:         for c in p.cells():
        ....:             for d in p.cells():
        ....:                 if P.lt(c, d) != Q.lt(d, c):
        ....:                     return False
        ....:     return True
        sage: all( test_duality(n, "NW", "SE") for n in range(7) )
        True
        sage: all( test_duality(n, "NE", "SW") for n in range(7) )
        True
        sage: all( test_duality(n, "NE", "SE") for n in range(4) )
        False

  `cells`()

    Return the coordinates of the cells of `self`. Coordinates are given as `(row-index, column-index)` and are 0 based.

    EXAMPLES:

        sage: SkewPartition([[4, 3, 1], [2]]).cells()
        [(0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0)]
        sage: SkewPartition([[4, 3, 1], []]).cells()
        [(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (2, 0)]
        sage: SkewPartition([[2], []]).cells()
        [(0, 0), (0, 1)]

  `column_lengths`()

    Return the column lengths of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[1,1]]).column_lengths()
        [1, 2, 1]
        sage: SkewPartition([[5,2,2,2],[2,1]]).column_lengths()
        [2, 3, 1, 1, 1]

  `columns_intersection_set`()

    Return the set of cells in the columns of the outer shape of `self` which columns intersect the skew diagram of `self`.

    EXAMPLES:

        sage: skp = SkewPartition([[3,2,1],[2,1]])
        sage: cells = Set([ (0,0), (0, 1), (0,2), (1, 0), (1, 1), (2, 0)])
        sage: skp.columns_intersection_set() == cells
        True

  `conjugate`()

    Return the conjugate of the skew partition skp.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[2]]).conjugate()
        [3, 2, 1] / [1, 1]

  `diagram`()

    Return the Ferrers diagram of `self`.

    EXAMPLES:

        sage: print SkewPartition([[5,4,3,1],[3,3,1]]).ferrers_diagram()
           **
           *
         **
        *
        sage: print SkewPartition([[5,4,3,1],[3,1]]).diagram()
           **
         ***
        ***
        *
        sage: SkewPartitions.global_options(diagram_str='#', convention="French")
        sage: print SkewPartition([[5,4,3,1],[3,1]]).diagram()
        #
        ###
         ###
           ##
        sage: SkewPartitions.global_options.reset()

  `ferrers_diagram`()

    Return the Ferrers diagram of `self`.

    EXAMPLES:

        sage: print SkewPartition([[5,4,3,1],[3,3,1]]).ferrers_diagram()
           **
           *
         **
        *
        sage: print SkewPartition([[5,4,3,1],[3,1]]).diagram()
           **
         ***
        ***
        *
        sage: SkewPartitions.global_options(diagram_str='#', convention="French")
        sage: print SkewPartition([[5,4,3,1],[3,1]]).diagram()
        #
        ###
         ###
           ##
        sage: SkewPartitions.global_options.reset()

  `frobenius_rank`()

    Return the Frobenius rank of the skew partition `self`.

    The Frobenius rank of a skew partition \(\lambda / \mu\) can be defined in various ways. The quickest one is probably the following: Writing \(\lambda\) as \((\lambda_1, \lambda_2, \cdots , \lambda_N)\), and writing \(\mu\) as \((\mu_1, \mu_2, \cdots , \mu_N)\), we define the Frobenius rank of \(\lambda / \mu\) to be the number of all \(1 \leq i \leq N\) such that

    \[\lambda_i - i \not\in \{ \mu_1 - 1, \mu_2 - 2, \cdots , \mu_N - N \}.\]

    In other words, the Frobenius rank of \(\lambda / \mu\) is the number of rows in the Jacobi-Trudi matrix of \(\lambda / \mu\) which don’t contain \(h_0\). Further definitions have been considered in [Stan2002] (where Frobenius rank is just being called rank).

    If \(\mu\) is the empty shape, then the Frobenius rank of \(\lambda / \mu\) is just the usual Frobenius rank of the partition \(\lambda\) (see `frobenius_rank()`).

    REFERENCES:

    [Stan2002] | Richard P. Stanley, The rank and minimal border strip decompositions of a skew partition, J. Combin. Theory Ser. A 100 (2002), pp. 349-375. Arxiv math/0109092v1.

    EXAMPLES:

        sage: SkewPartition([[8,8,7,4], [4,1,1]]).frobenius_rank()
        4
        sage: SkewPartition([[2,1], [1]]).frobenius_rank()
        2
        sage: SkewPartition([[2,1,1], [1]]).frobenius_rank()
        2
        sage: SkewPartition([[2,1,1], [1,1]]).frobenius_rank()
        2
        sage: SkewPartition([[5,4,3,2], [2,1,1]]).frobenius_rank()
        3
        sage: SkewPartition([[4,2,1], [3,1,1]]).frobenius_rank()
        2
        sage: SkewPartition([[4,2,1], [3,2,1]]).frobenius_rank()
        1

    If the inner shape is empty, then the Frobenius rank of the skew partition is just the standard Frobenius rank of the partition:

        sage: all( SkewPartition([lam, Partition([])]).frobenius_rank()
        ....:      == lam.frobenius_rank() for i in range(6)
        ....:      for lam in Partitions(i) )
        True

    If the inner and outer shapes are equal, then the Frobenius rank is zero:

        sage: all( SkewPartition([lam, lam]).frobenius_rank() == 0
        ....:      for i in range(6) for lam in Partitions(i) )
        True

  `inner`()

    Return the inner partition of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[1,1]]).inner()
        [1, 1]

  `inner_corners`()

    Return a list of the inner corners of `self`.

    EXAMPLES:

        sage: SkewPartition([[4, 3, 1], [2]]).inner_corners()
        [(0, 2), (1, 0)]
        sage: SkewPartition([[4, 3, 1], []]).inner_corners()
        [(0, 0)]

  `is_connected`()

    Return `True` if `self` is a connected skew partition.

    A skew partition is said to be connected if for each pair of consecutive rows, there are at least two cells (one in each row) which have a common edge.

    EXAMPLES:

        sage: SkewPartition([[5,4,3,1],[3,3,1]]).is_connected()
        False
        sage: SkewPartition([[5,4,3,1],[3,1]]).is_connected()
        True

  `is_overlap`(n)

    Return `True` if the overlap of `self` is at most `n`.

    See also

    `overlap()`

    EXAMPLES:

        sage: SkewPartition([[5,4,3,1],[3,1]]).is_overlap(1)
        True

  `is_ribbon`()

    Return `True` if and only if `self` is a ribbon, that is if it has no \(2 \times 2\) boxes.

    EXAMPLES:

        sage: SkewPartition([[3,3,1],[2,2]]).is_ribbon()
        True
        sage: SkewPartition([[3,1],[3]]).is_ribbon()
        True
        sage: SkewPartition([[3,3,2],[2]]).is_ribbon()
        False

  `jacobi_trudi`()

    Return the Jacobi-Trudi matrix of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[2,1]]).jacobi_trudi()
        [h[1]    0    0]
        [h[3] h[1]    0]
        [h[5] h[3] h[1]]
        sage: SkewPartition([[4,3,2],[2,1]]).jacobi_trudi()
        [h[2]  h[]    0]
        [h[4] h[2]  h[]]
        [h[6] h[4] h[2]]

  `k_conjugate`(k)

    Return the \(k\)-conjugate of the skew partition.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[2,1]]).k_conjugate(3)
        [2, 1, 1, 1, 1] / [2, 1]
        sage: SkewPartition([[3,2,1],[2,1]]).k_conjugate(4)
        [2, 2, 1, 1] / [2, 1]
        sage: SkewPartition([[3,2,1],[2,1]]).k_conjugate(5)
        [3, 2, 1] / [2, 1]

  `outer`()

    Return the outer partition of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[1,1]]).outer()
        [3, 2, 1]

  `outer_corners`()

    Return a list of the outer corners of `self`.

    EXAMPLES:

        sage: SkewPartition([[4, 3, 1], [2]]).outer_corners()
        [(0, 3), (1, 2), (2, 0)]

  `overlap`()

    Return the overlap of `self`.

    The overlap of two consecutive rows in a skew partition is the number of pairs of cells (one in each row) that share a common edge. This number can be positive, zero, or negative.

    The overlap of a skew partition is the minimum of the overlap of the consecutive rows, or infinity in the case of at most one row. If the overlap is positive, then the skew partition is called connected.

    EXAMPLES:

        sage: SkewPartition([[],[]]).overlap()
        +Infinity
        sage: SkewPartition([[1],[]]).overlap()
        +Infinity
        sage: SkewPartition([[10],[]]).overlap()
        +Infinity
        sage: SkewPartition([[10],[2]]).overlap()
        +Infinity
        sage: SkewPartition([[10,1],[2]]).overlap()
        -1
        sage: SkewPartition([[10,10],[1]]).overlap()
        9

  `pieri_macdonald_coeffs`()

    Computation of the coefficients which appear in the Pieri formula for Macdonald polynomials given in his book ( Chapter 6.6 formula 6.24(ii) )

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[2,1]]).pieri_macdonald_coeffs()
        1
        sage: SkewPartition([[3,2,1],[2,2]]).pieri_macdonald_coeffs()
        (q^2*t^3 - q^2*t - t^2 + 1)/(q^2*t^3 - q*t^2 - q*t + 1)
        sage: SkewPartition([[3,2,1],[2,2,1]]).pieri_macdonald_coeffs()
        (q^6*t^8 - q^6*t^6 - q^4*t^7 - q^5*t^5 + q^4*t^5 - q^3*t^6 + q^5*t^3 + 2*q^3*t^4 + q*t^5 - q^3*t^2 + q^2*t^3 - q*t^3 - q^2*t - t^2 + 1)/(q^6*t^8 - q^5*t^7 - q^5*t^6 - q^4*t^6 + q^3*t^5 + 2*q^3*t^4 + q^3*t^3 - q^2*t^2 - q*t^2 - q*t + 1)
        sage: SkewPartition([[3,3,2,2],[3,2,2,1]]).pieri_macdonald_coeffs()
        (q^5*t^6 - q^5*t^5 + q^4*t^6 - q^4*t^5 - q^4*t^3 + q^4*t^2 - q^3*t^3 - q^2*t^4 + q^3*t^2 + q^2*t^3 - q*t^4 + q*t^3 + q*t - q + t - 1)/(q^5*t^6 - q^4*t^5 - q^3*t^4 - q^3*t^3 + q^2*t^3 + q^2*t^2 + q*t - 1)

  `pp`()

    Pretty-print `self`.

    EXAMPLES:

        sage: SkewPartition([[5,4,3,1],[3,3,1]]).pp()
           **
           *
         **
        *

  `quotient`(k)

    The quotient map extended to skew partitions.

    EXAMPLES:

        sage: SkewPartition([[3, 3, 2, 1], [2, 1]]).quotient(2)
        [[3] / [], [] / []]

  `row_lengths`()

    Return the row lengths of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[1,1]]).row_lengths()
        [2, 1, 1]

  `rows_intersection_set`()

    Return the set of cells in the rows of the outer shape of `self` which rows intersect the skew diagram of `self`.

    EXAMPLES:

        sage: skp = SkewPartition([[3,2,1],[2,1]])
        sage: cells = Set([ (0,0), (0, 1), (0,2), (1, 0), (1, 1), (2, 0)])
        sage: skp.rows_intersection_set() == cells
        True

  `size`()

    Return the size of `self`.

    EXAMPLES:

        sage: SkewPartition([[3,2,1],[1,1]]).size()
        4

  `to_dag`()

    Return a directed acyclic graph corresponding to the skew partition `self`.

    The directed acyclic graph corresponding to a skew partition \(p\) is the digraph whose vertices are the cells of \(p\), and whose edges go from each cell to its lower and right neighbors (in English notation).

    EXAMPLES:

        sage: dag = SkewPartition([[3, 2, 1], [1, 1]]).to_dag()
        sage: dag.edges()
        [('0,1', '0,2', None), ('0,1', '1,1', None)]
        sage: dag.vertices()
        ['0,1', '0,2', '1,1', '2,0']

  `to_list`()

    Return `self` as a list of lists.

    EXAMPLES:

        sage: s = SkewPartition([[4,3,1],[2]])
        sage: s.to_list()
        [[4, 3, 1], [2]]
        sage: type(s.to_list())
        <type 'list'>

class `sage.combinat.skew_partition.``SkewPartitions`(is_infinite=False)

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

  Skew partitions.

  Warning

  The iterator of this class only yields skew partitions which are reduced, in the sense that there are no empty rows before the last nonempty row, and there are no empty columns before the last nonempty column.

  EXAMPLES:

      sage: SkewPartitions(4)
      Skew partitions of 4
      sage: SkewPartitions(4).cardinality()
      28
      sage: SkewPartitions(row_lengths=[2,1,2])
      Skew partitions with row lengths [2, 1, 2]
      sage: SkewPartitions(4, overlap=2)
      Skew partitions of 4 with a minimum overlap of 2
      sage: SkewPartitions(4, overlap=2).list()
      [[4] / [], [2, 2] / []]

  `Element`

    alias of `SkewPartition`

  `from_row_and_column_length`(rowL, colL)

    Construct a partition from its row lengths and column lengths.

    INPUT:

    - `rowL` – A composition or a list of positive integers
    - `colL` – A composition or a list of positive integers

    OUTPUT:

    - If it exists the unique skew-partitions with row lengths `rowL` and column lengths `colL`.
    - Raise a `ValueError` if `rowL` and `colL` are not compatible.

    EXAMPLES:

        sage: S = SkewPartitions()
        sage: print S.from_row_and_column_length([3,1,2,2],[2,3,1,1,1]).diagram()
          ***
         *
        **
        **
        sage: S.from_row_and_column_length([],[])
        [] / []
        sage: S.from_row_and_column_length([1],[1])
        [1] / []
        sage: S.from_row_and_column_length([2,1],[2,1])
        [2, 1] / []
        sage: S.from_row_and_column_length([1,2],[1,2])
        [2, 2] / [1]
        sage: S.from_row_and_column_length([1,2],[1,3])
        Traceback (most recent call last):
        ...
        ValueError: Sum mismatch : [1, 2] and [1, 3]
        sage: S.from_row_and_column_length([3,2,1,2],[2,3,1,1,1])
        Traceback (most recent call last):
        ...
        ValueError: Incompatible row and column length : [3, 2, 1, 2] and [2, 3, 1, 1, 1]

    Warning

    If some rows and columns have length zero, there is no way to retrieve unambiguously the skew partition. We therefore raise a `ValueError`. For examples here are two skew partitions with the same row and column lengths:

        sage: skp1 = SkewPartition([[2,2],[2,2]])
        sage: skp2 = SkewPartition([[2,1],[2,1]])
        sage: skp1.row_lengths(), skp1.column_lengths()
        ([0, 0], [0, 0])
        sage: skp2.row_lengths(), skp2.column_lengths()
        ([0, 0], [0, 0])
        sage: SkewPartitions().from_row_and_column_length([0,0], [0,0])
        Traceback (most recent call last):
        ...
        ValueError: row and column length must be positive

    TESTS:

        sage: all(SkewPartitions().from_row_and_column_length(p.row_lengths(), p.column_lengths()) == p
        ....:       for i in range(8) for p in SkewPartitions(i))
        True

  `global_options`(*get_value, **set_value)

    Sets and displays the global options for elements of the skew partition classes. If no parameters are set, then the function returns a copy of the options dictionary.

    The `options` to skew partitions can be accessed as the method `SkewPartitions.global_options` of `SkewPartitions` and related parent classes.

    OPTIONS:

    - `convention` – (default: `English`) Sets the convention used for displaying tableaux and partitions
      - `English` – use the English convention
      - `French` – use the French convention
    - `diagram_str` – (default: `*`) The character used for the cells when printing Ferrers diagrams
    - `display` – (default: `quotient`) Specifies how skew partitions should be printed
      - `array` – alias for `diagram`
      - `diagram` – as a skew Ferrers diagram
      - `ferrers_diagram` – alias for `diagram`
      - `lists` – displayed as a pair of lists
      - `pair` – alias for `lists`
      - `quotient` – displayed as a quotient of partitions
      - `young_diagram` – alias for `diagram`
    - `latex` – (default: `young_diagram`) Specifies how skew partitions should be latexed
      - `array` – alias for `diagram`
      - `diagram` – latex as a skew Ferrers diagram
      - `ferrers_diagram` – alias for `diagram`
      - `marked` – latex as a partition where the skew shape is marked
      - `young_diagram` – latex as a skew Young diagram
    - `latex_diagram_str` – (default: `\ast`) The character used for the cells when latexing Ferrers diagrams
    - `latex_marking_str` – (default: `X`) The character used to marked the deleted cells when latexing marked partitions
    - `notation` – alternative name for `convention`

    EXAMPLES:

        sage: SP = SkewPartition([[4,2,2,1], [3, 1, 1]])
        sage: SP
        [4, 2, 2, 1] / [3, 1, 1]
        sage: SkewPartitions.global_options(display="lists")
        sage: SP
        [[4, 2, 2, 1], [3, 1, 1]]

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

        sage: SkewPartitions.global_options(display="diagram", convention='French')
        sage: SP
        *
         *
         *
           *
        sage: T = Tableau([[1,2,3],[4,5]])
        sage: T.pp()
          4  5
          1  2  3
        sage: P = Partition([4, 2, 2, 1])
        sage: P.pp()
        *
        **
        **
        ****
        sage: Tableaux.global_options(convention="english")
        sage: SP
           *
         *
         *
        *
        sage: T.pp()
          1  2  3
          4  5
        sage: SkewPartitions.global_options.reset()

    See `GlobalOptions` for more features of these options.

class `sage.combinat.skew_partition.``SkewPartitions_all`

  Bases: `sage.combinat.skew_partition.SkewPartitions`

  Class of all skew partitions.

class `sage.combinat.skew_partition.``SkewPartitions_n`(n, overlap)

  Bases: `sage.combinat.skew_partition.SkewPartitions`

  The set of skew partitions of `n` with overlap at least `overlap` and no empty row.

  INPUT:

  - `n` – a non-negative integer
  - `overlap` – an integer (default: \(0\))

  Caveat: this set is stable under conjugation only for `overlap` equal to 0 or 1. What exactly happens for negative overlaps is not yet well specified and subject to change (we may want to introduce vertical overlap constraints as well).

  Todo

  As is, this set is essentially the composition of `Compositions(n)` (which give the row lengths) and `SkewPartition(n, row_lengths=...)`, and one would want to “inherit” list and cardinality from this composition.

  `cardinality`()

    Return the number of skew partitions of the integer \(n\) (with given overlap, if specified; and with no empty rows before the last row).

    EXAMPLES:

        sage: SkewPartitions(0).cardinality()
        1
        sage: SkewPartitions(4).cardinality()
        28
        sage: SkewPartitions(5).cardinality()
        87
        sage: SkewPartitions(4, overlap=1).cardinality()
        9
        sage: SkewPartitions(5, overlap=1).cardinality()
        20
        sage: s = SkewPartitions(5, overlap=-1)
        sage: s.cardinality() == len(s.list())
        True

class `sage.combinat.skew_partition.``SkewPartitions_rowlengths`(co, overlap)

  Bases: `sage.combinat.skew_partition.SkewPartitions`

  All skew partitions with given row lengths.

`sage.combinat.skew_partition.``from_row_and_column_length`(rowL, colL)

  This has been deprecated in trac ticket #14101. Use `SkewPartitions().from_row_and_column_length()` instead.

  EXAMPLES:

      sage: sage.combinat.skew_partition.from_row_and_column_length([3,1,2,2],[2,3,1,1,1])
      doctest:...: DeprecationWarning: from_row_and_column_length is deprecated. Use SkewPartitions().from_row_and_column_length instead.
      See http://trac.sagemath.org/14101 for details.
      [5, 2, 2, 2] / [2, 1]

`sage.combinat.skew_partition.``row_lengths_aux`(skp)

  EXAMPLES:

      sage: from sage.combinat.skew_partition import row_lengths_aux
      sage: row_lengths_aux([[5,4,3,1],[3,3,1]])
      [2, 1, 2]
      sage: row_lengths_aux([[5,4,3,1],[3,1]])
      [2, 3]

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