KostkaJackNumbersWithGivenLambda {jack}

Kostka-Jack numbers with a given partition \lambda

Description

Kostka numbers with Jack parameter, or Kostka-Jack numbers K_{\lambda,\mu}(\alpha) for a given Jack parameter \alpha and a given integer partition \lambda.

Usage

KostkaJackNumbersWithGivenLambda(lambda, alpha, output = "vector")

Arguments

lambda

integer partition

alpha

the Jack parameter, a bigq number or anything coercible to a bigq number

output

the format of the output, either "vector" or "list"

Details

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

Value

If output="vector", this function returns a named vector. This vector is made of the non-zero (i.e. positive) Kostka-Jack numbers K_{\lambda,\mu}(\alpha) given as character strings and its names encode the partitions \mu. If ouput="list", this function returns a list of lists. Each of these lists has two elements. The first one is named mu and is an integer partition, and the second one is named value and is a bigq rational number, the Kostka-Jack number K_{\lambda,\mu}(\alpha).

See Also

KostkaJackNumbers, symbolicKostkaJackNumbersWithGivenLambda.

Examples

KostkaJackNumbersWithGivenLambda(c(3, 2), alpha = "2")