Skew Kostka-Jack numbers associated to a given skew partition and a given Jack parameter.
skewKostkaJackNumbers(lambda, mu, alpha = NULL, output = "vector")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.
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)
the Jack parameter, a bigq number or an object coercible
to a bigq number; setting alpha=NULL is equivalent to set
alpha=1
the format of the output, either "vector" or
"list"
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.
symbolicSkewKostkaJackNumbers.
skewKostkaJackNumbers(c(4,2,2), c(2,2))
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