KostkaJackNumbersWithGivenLambda function

Kostka-Jack numbers with a given partition $\lambda$

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

and a given integer partition $\lambda$.

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"

Returns

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 $K_{\lambda,\mu}(\alpha)$.

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.

Examples

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

See Also

KostkaJackNumbers, symbolicKostkaJackNumbersWithGivenLambda.