I'm looking for information about how representations of $S_n$ decompose under restriction.

I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S_n$ decompose into a direct sum of irreducible modules of $S_{n-1}$ and there is a nice combinatorial rule for determining the multiplicities of the $S_{n-1}$-modules in the decomposition.

Are there similar results for the embedding of $S_k \times S_{n-k}$ in $S_n$? That is, is there a way of computing the multiplicity of the irreducible $S_k \times S_{n-k}$-module $L(\nu) \otimes L(\mu)$ in $L(\lambda) \downarrow^{S_n}_{S_k \times S_{n-k}}$?

Young subgroups; inducing or restricting characters from or to them is a classical theme in the subject, going back I presume to Young himself. $\endgroup$