combinat::skewPartitions – skew partitions

The library combinat::skewPartitions provides functions for counting, generating, and manipulating skew partitions of fixed size.

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Superdomain

Dom::BaseDomain

Categories

Cat::CombinatorialClass

Axioms

Ax::systemRep

Related Domains:

combinat::partitions

Details:

• A skew partition  of size is a pair of two partitions where is a partition of the integer , is a partition of the integer , is an inner partition of and . We say that and are respectively the outer and inner partitions of .

• A skew partition is so represented in MuPAD by a list of two partitions:

  [outer, inner].

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

  
    +---+
    |   |
    +---+---+---+
        |   |   |
        +---+---+---+
                |   |
                +---+---+
                |   |   |
                +---+---+
  

  And this is not a skew partition:

  
    +---+  
    |   |
    +---+---+---+
        |   |   |  
        +---+---+
  
                +---+---+
                |   |   |  
                +---+---+
  

   

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

  
    +---+  
    |   |
    +---+---+---+
        |   |   |  
        +---+---+---+
        |   |   |   |  
        +---+---+---+---+
                |   |   |  
                +---+---+
  

  And our first example of skew partition is not a connected one.

• To apply a reflection w.r.t. the main diagonal to the diagram yields the diagram of the conjugate skew partition, here [[4,3,3,2,1],[3,2,2]]:

  
    +---+
    |   |
    +---+---+---+
        |   |   |
        +---+---+---+
                |   |    
                +---+
                |   |
                +---+---+
                    |   |
                    +---+
  

  See combinat::skewPartitions::conjugate.

• 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. See the examples below.

Entries

isA – test if a object is a skew partition

combinat::skewPartitions::isA(any type , <nonnegative integer n, <Connected>>)

Tests whether skp is a skew partition. The second argument is either Connectedor a non negative integer. In the case it is a non negative integer, it indicates the size of the skew partition and the function tests if skp is a skew partition of size n, and if it is Connectedthe function tests whether skp is a connected skew partition. With three arguments it is possible to tests if skp is a connected skew partition of size n.

count – number of skew partitions

combinat::skewPartitions::count(nonnegative integer , <Connected>)

Returns the number of skew partitions of size n. The result counts only the connected skew partitions if it is said so.

generator – generator for skew partitions

combinat::skewPartitions::generator(nonnegative integer , <Connected>)

Returns a generator for the skew partitions of size n. The result gives access only to connected skew partitions if it is said so.

list – list of skew partitions

combinat::skewPartitions::list(nonnegative integer , <Connected>)

Returns the list of the skew partitions of size n. The result is a list of connected skew partitions if it is said so.

conjugate – conjugate of a skew partition

combinat::skewPartitions::conjugate(skew partition skp)

Returns the conjugate (or transposed) of the skew partition skp. Geometrically, this operation amounts to a reflection of the diagram of the partition with respect to the main diagonal.

rowLengths – row-lengths of skew partition

combinat::skewPartitions::rowLengths(skew partition skp)

Returns the row-lengths that is the list of the numbers of boxes in each rows of the skew partition skp, starting with the bottom row.

columnLengths – column-lengths of skew partition

combinat::skewPartitions::columnLengths(skew partition skp)

Returns the column-lengths that is the list of the numbers of boxes in each columns of the skew partition skp.

fromRowAndColumnLengths – retrieve a skew partition from it row and column-lengths

combinat::skewPartitions::fromRowAndColumnLengths(composition rowL, composition colL)

Returns the skew-partitions whose row and column-lengths are rowL and colL, returns FAIL if there is no such skew-partitions.

jacobiTrudy – Jacoby-Trudi formula

combinat::skewPartitions::jacobiTrudy(skew partition , <function f>, <domain coeffRing>)

This function compute the Jacobi-Trudy matrix associated to the skew partition skp. Its determinant express the skew Schur polynomial associated to skp on the basis of homogenous complete symmetric polynomial (see Macdonald I.-G., (1995), "Symmetric Functions and Hall Polynomials", Oxford Science Publication). By default the function argument is i -> h[i] and coeffRing is Dom::ExpressionField() . These two options permits to obtain matrix entries as true symmetric functions.

innerCorners – inner corners of a skew partition

combinat::skewPartitions::innerCorners(skew partition skp)

Returns the list of coordinates of the inner corner boxes in the diagram of the skew partition.

outerCorners – outer corners of a skew partition

combinat::skewPartitions::outerCorners(skew partition skp)

Returns the list of coordinates of the outer corner boxes in the diagram of the skew partition, i.e. the corners of the outer partition of skp.

printPretty – graphic representation of a skew partition

combinat::skewPartitions::printPretty(skew partition skp)

Returns a graphic representation of the skew partition as rows of boxes.

printCompact – graphic representation of a skew partition

combinat::skewPartitions::printCompact(skew partition skp)

Same as above, but in a more compact way, where each box is represented by #.

printPrettyCorners – graphic representation of inner and outer corners of a skew partition

combinat::skewPartitions::printPrettyCorners(skew partition skp)

Returns a graphic representation of the skew partition as rows of boxes where inner corners are marked with x and outer corners are marked with *. A % marks a corner which is inner and outer at the same time.

printCompactCorners – graphic representation of inner and outer corners of a skew partition

combinat::skewPartitions::printCompactCorners(skew partition skp)

Same as above, but in a more compact way, where each box is represented by #, and x, *, % mark the corners.

Example 1:

Example 2:

Example 3:

Example 4: