Skew Partitions

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

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

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

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

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

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

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

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

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

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

EXAMPLES:

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

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

There are 4 connected skew partitions of size 3:

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

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

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

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

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

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

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

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

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

REFERENCES:

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

AUTHORS:

  • Mike Hansen: Initial version
  • Travis Scrimshaw (2013-02-11): Factored out CombinatorialClass
class sage.combinat.skew_partition.SkewPartition(parent, skp)

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

A skew partition.

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

cells()

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

EXAMPLES:

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

Return the column lengths of self.

EXAMPLES:

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

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

EXAMPLES:

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

Return the conjugate of the skew partition skp.

EXAMPLES:

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

Return the Ferrers diagram of self.

EXAMPLES:

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

Return the Ferrers diagram of self.

EXAMPLES:

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

Return the Frobenius rank of the skew partition self.

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

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

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

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

REFERENCES:

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

EXAMPLES:

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

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

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

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

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

Return the inner partition of self.

EXAMPLES:

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

Return a list of the inner corners of self.

EXAMPLES:

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

Return True if self is a connected skew partition.

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

EXAMPLES:

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

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

See also

overlap()

EXAMPLES:

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

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

EXAMPLES:

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

Return the Jacobi-Trudi matrix of self.

EXAMPLES:

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

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

EXAMPLES:

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

Return the outer partition of self.

EXAMPLES:

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

Return a list of the outer corners of self.

EXAMPLES:

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

Return the overlap of self.

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

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

EXAMPLES:

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

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

EXAMPLES:

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

Pretty-print self.

EXAMPLES:

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

The quotient map extended to skew partitions.

EXAMPLES:

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

Return the row lengths of self.

EXAMPLES:

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

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

EXAMPLES:

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

Return the size of self.

EXAMPLES:

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

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

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

EXAMPLES:

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

Return self as a list of lists.

EXAMPLES:

sage: s = SkewPartition([[4,3,1],[2]])
sage: s.to_list()
[[4, 3, 1], [2]]
sage: type(s.to_list())
<type 'list'>
class sage.combinat.skew_partition.SkewPartitions(is_infinite=False)

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

Skew partitions.

EXAMPLES:

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

alias of SkewPartition

from_row_and_column_length(rowL, colL)

Construct a partition from its row lengths and column lengths.

INPUT:

  • rowL – A composition or a list of positive integers
  • colL – A composition or a list of positive integers

OUTPUT:

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

EXAMPLES:

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

Warning

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

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

TESTS:

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

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

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

OPTIONS:

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

EXAMPLES:

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

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

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

See GlobalOptions for more features of these options.

class sage.combinat.skew_partition.SkewPartitions_all

Bases: sage.combinat.skew_partition.SkewPartitions

Class of all skew partitions.

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

Bases: sage.combinat.skew_partition.SkewPartitions

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

INPUT:

  • n – A non-negative integer
  • overlap – An integer

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 \(n\).

EXAMPLES:

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

Bases: sage.combinat.skew_partition.SkewPartitions

All skew partitions with given row lengths.

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

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

EXAMPLES:

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

EXAMPLES:

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

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