kostkaNumber -- Compute a Kostka number

The Kostka number K_{\lambda,\mu} is the number of semistandard Young tableaux of shape \lambda and content \mu. Equivalently, it is the coefficient of the Schur function s_\lambda in the product of complete symmetric functions h_{\mu_1} h_{\mu_2} \cdots.

Returns 0 whenever \lambda and \mu do not have the same size, or whenever \mu does not dominate \lambda (so that no tableaux of the requested shape and content exist).

The full Kostka matrix K_{\lambda,\mu} for partitions of 4, listed in reverse dominance order (4), (3,1), (2,2), (2,1,1), (1,1,1,1), is upper triangular with ones on the diagonal (a reflection of the fact that Schur functions form an h-triangular basis with respect to the monomial basis):

A useful boundary identity: for any partition \mu of n, the Kostka number with \lambda = (n) is 1, counting the unique row tableau:

Similarly, the Kostka number K_{\lambda,(1^n)} counts standard Young tableaux of shape \lambda, which for \lambda = (2,2) is 2 (the hook-length value), matching the dimension of the corresponding irreducible S_4-representation: