Kostka multiplicity one for multipartitions

If $[\lambda(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $\mu$ is a partition of $n$, we study the number $K_{[\lambda(j)]\mu}$ of sequences of semistandard Young tableaux of shape $[\lambda(j)]$ and total weight $\mu$. We show that the numbers $K_{[\lambda(j)] \mu}$ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of conditions on $[\lambda(j)]$ and $\mu$ which are equivalent to $K_{[\lambda(j)] \mu} = 1$, generalizing a theorem of Berenshte\u{\i}n and Zelevinski\u{\i}. We also show that the questions of whether $K_{[\lambda(j)] \mu} > 0$ or $K_{[\lambda(j)] \mu} = 1$ can be answered in polynomial time, expanding on a result of Narayanan. Finally, we give an application to multiplicities in the degenerate Gel'fand-Graev representations of the finite general linear group, and we show that the problem of determining whether a given irreducible representation of the finite general linear group appears with nonzero multiplicity in a given degenerate Gel'fand-Graev representation, with their partition parameters as input, is $NP$-complete.