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 [Mac1995] for a definition and discussion.

sage: SkewPartition([[4,3,1],[2]]).jacobi_trudi()                                   # needs sage.modules
[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)]

AUTHORS:

Mike Hansen: Initial version
Travis Scrimshaw (2013-02-11): Factored out CombinatorialClass
Trevor K. Karn (2022-08-03): Add outside_corners

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

Bases: CombinatorialElement

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: # needs sage.graphs
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: # needs sage.graphs
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: # needs sage.graphs
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]

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.options(diagram_str='#', convention="French")
sage: print(SkewPartition([[5,4,3,1],[3,1]]).diagram())
#
###
 ###
   ##
sage: SkewPartitions.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.options(diagram_str='#', convention="French")
sage: print(SkewPartition([[5,4,3,1],[3,1]]).diagram())
#
###
 ###
   ##
sage: SkewPartitions.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 [Sta2002] (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()).

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

EXAMPLES:

sage: mu = SkewPartition([[3,2,1],[2,1]])
sage: mu.pp()
  *
 *
*
sage: mu.outside_corners()
[(0, 3), (1, 2), (2, 1), (3, 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

specht_module(base_ring=None)#

Return the Specht module corresponding to self.

EXAMPLES:

sage: mu = SkewPartition([[3,2,1], [2]])
sage: SM = mu.specht_module(QQ)                                             # needs sage.modules
sage: s = SymmetricFunctions(QQ).s()                                        # needs sage.modules
sage: s(SM.frobenius_image())                                               # needs sage.modules
s[2, 1, 1] + s[2, 2] + s[3, 1]

We verify that the Frobenius image is the corresponding skew Schur function:

sage: s[3,2,1].skew_by(s[2])                                                # needs sage.modules
s[2, 1, 1] + s[2, 2] + s[3, 1]

sage: mu = SkewPartition([[4,2,1], [2,1]])
sage: SM = mu.specht_module(QQ)                                             # needs sage.modules
sage: s(SM.frobenius_image())                                               # needs sage.modules
s[2, 1, 1] + s[2, 2] + 2*s[3, 1] + s[4]
sage: s(mu)                                                                 # needs sage.modules
s[2, 1, 1] + s[2, 2] + 2*s[3, 1] + s[4]

specht_module_dimension(base_ring=None)#

Return the dimension of the Specht module corresponding to self.

This is equal to the number of standard (skew) tableaux of shape self.

EXAMPLES:

sage: mu = SkewPartition([[3,2,1], [2]])
sage: mu.specht_module_dimension()                                          # needs sage.modules
8
sage: mu.specht_module_dimension(GF(2))                                     # needs sage.modules sage.rings.finite_rings
8

to_dag(format='string')#

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).

INPUT:

format – either 'string' or 'tuple' (default: 'string'); determines whether the vertices of the resulting dag will be strings or 2-tuples of coordinates

EXAMPLES:

sage: # needs sage.graphs
sage: dag = SkewPartition([[3, 3, 1], [1, 1]]).to_dag()
sage: dag.edges(sort=True)
[('0,1', '0,2', None),
('0,1', '1,1', None),
('0,2', '1,2', None),
('1,1', '1,2', None)]
sage: dag.vertices(sort=True)
['0,1', '0,2', '1,1', '1,2', '2,0']
sage: dag = SkewPartition([[3, 2, 1], [1, 1]]).to_dag(format="tuple")
sage: dag.edges(sort=True)
[((0, 1), (0, 2), None), ((0, 1), (1, 1), None)]
sage: dag.vertices(sort=True)
[(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())
<class 'list'>

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

Bases: UniqueRepresentation, Parent

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

options = Current options for SkewPartitions - convention: English - diagram_str: * - display: quotient - latex: young_diagram - latex_diagram_str: \ast - latex_marking_str: X#

class sage.combinat.skew_partition.SkewPartitions_all#

Bases: SkewPartitions

Class of all skew partitions.

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

Bases: 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: SkewPartitions

All skew partitions with given row lengths.

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]