KostkaJackNumbers function

Kostka-Jack numbers with a given Jack parameter

Kostka numbers with Jack parameter, or Kostka-Jack numbers, for partitions of a given weight and a given Jack parameter.

KostkaJackNumbers(n, alpha = "1")

Arguments

• n: positive integer, the weight of the partitions
• alpha: the Jack parameter, a bigq number or an object coercible to a bigq number

Returns

The matrix of the Kostka-Jack numbers $K_{\lambda,\mu}(\alpha)$

given as character strings representing integers or fractions. The row names of this matrix encode the partitions $\lambda$ and the column names encode the partitions $\mu$

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

KostkaJackNumbers(4)

See Also

KostkaJackNumbersWithGivenLambda, symbolicKostkaJackNumbers, skewKostkaJackNumbers.