# 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] .

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)

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)


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)


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)


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)
{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)
{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()
{}
{}


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)
[(2, 0)]
[(3, 0), (0, 2)]
[(2, 1)]
[(3, 1), (0, 3)]
[(4, 0)]
[]


TESTS:

sage: StrongTableau([[None, None], [None]], 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.


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
2


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


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)
[-2]
[-3, 2]
[-1]
[-2, 3]
[-4]
[]


TESTS:

sage: StrongTableau([[None, None], [None]], 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.

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.

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.

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 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)

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

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]]
[]

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]]


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]])
{}
{}

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 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: StrongTableaux.marked_CST_to_transposition_sequence([[-1, -1, -1], [1]], 2)
[[2, 3], [1, 2], [0, 1]]
sage: StrongTableaux.marked_CST_to_transposition_sequence([], 2)
[]
sage: StrongTableaux.marked_CST_to_transposition_sequence([[-2, -2, -2], [2]], 2)
[[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

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.

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)

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)

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].

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

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

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)

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

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)

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)

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)

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
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)


TableauTuples

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Symmetric Functions