Let $\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$ and $\mu = (\mu_1 \geq \mu_2 \geq \cdots \geq \mu_\ell)$ be partitions of a positive integer $n$. As in Fulton's book on Young tableaux, recall that a tabloid is an equivalence class of numberings (with the distinct numbers $1,2,\ldots,n$) of a Young diagram, two being equivalent if corresponding rows contain the same entries. The symmetric group $S_n$ then acts in a natural way on tabloids. Let $M^\lambda$ denote the complex vector space with basis the tabloids of shape $\lambda$. Since $S_n$ acts on tabloids, $M^\lambda$ is naturally an $S_n$-representation.

Now let $S_\mu$ denote the Young subgroup of $S_n$ corresponding to the partition $\mu$. For example if $n=7$ and $\mu=(3,2,2)$ then $S_\mu = S_{\{1,2,3\}} \times S_{\{4,5\}} \times S_{\{6,7\}} \cong S_3 \times S_2 \times S_2$.

I am interested in computing the dimension of the $S_\mu$-invariant subspace of $M^\lambda$, i.e. $dim (M^\lambda)^{S_\mu}$. This amounts to counting the number of orbits under $S_\mu$ in the set of tabloids of shape $\lambda$.

A simple reformulation in terms of ``balls and boxes'' would be: with $\lambda$ and $\mu$ the fixed partitions as above, I have $n$ balls in $\ell$ different colors, where there are $\mu_i$ many balls of the $i$-th color. I want to count the number of ways to distribute these balls into $k$ many boxes, where the $j$-th box can hold at most $\lambda_j$ many balls.

Does anyone know a formula for this number? Alternatively, any concrete information about these numbers would be appreciated.