skewKostkaJackNumbers: Skew Kostka-Jack numbers with given Jack parameter

In jack: Jack, Zonal, Schur, and Other Symmetric Polynomials

Skew Kostka-Jack numbers with given Jack parameter

Description

Skew Kostka-Jack numbers associated to a given skew partition and a given Jack parameter.

Usage

skewKostkaJackNumbers(lambda, mu, alpha = NULL, output = "vector")

Arguments

lambda, mu	integer partitions defining the skew partition: lambda is the outer partition and mu is the inner partition (so mu must be a subpartition of lambda)
alpha	the Jack parameter, a bigq number or an object coercible to a bigq number; setting alpha=NULL is equivalent to set alpha=1
output	the format of the output, either "vector" or "list"

Details

The skew Kostka-Jack number K_{\lambda/\mu,\nu}(\alpha) is the coefficient of the monomial symmetric polynomial m_\nu in the expression of the skew P-Jack polynomial P_{\lambda/\mu}(\alpha) as a linear combination of monomial symmetric polynomials. For \alpha=1 it is the ordinary skew Kostka number.

Value

If output="vector", the function returns a named vector. This vector is made of the non-zero skew Kostka-Jack numbers K_{\lambda/\mu,\nu}(\alpha) given as character strings and its names encode the partitions \nu. If ouput="list", the function returns a list. Each element of this list is a named list with two elements: an integer partition \nu in the field named "nu", and the corresponding skew Kostka-Jack number K_{\lambda/\mu,\nu}(\alpha) in the field named "value". Only the non-null skew Kostka-Jack numbers are provided by this list.

Note

The skew Kostka-Jack numbers K_{\lambda/\mu,\nu}(\alpha) are well defined when the Jack parameter \alpha is zero, however this function does not work with alpha=0. A possible way to get the skew Kostka-Jack numbers K_{\lambda/\mu,\nu}(0) is to use the function symbolicSkewKostkaJackNumbers to get the skew Kostka-Jack numbers with a symbolic Jack parameter \alpha, and then to substitute \alpha with 0.

See Also

symbolicSkewKostkaJackNumbers.

Examples

skewKostkaJackNumbers(c(4,2,2), c(2,2))

jack documentation built on Sept. 11, 2024, 6:34 p.m.