Restriction coefficients for partitions with at most three columns

Let $r\ge 0$, and let $\lambda$ and $\mu$ be partitions such that ${\lambda }_1\le r+1$. We present a combinatorial interpretation of the plethysm coefficient $⟨s_{\lambda },s_{\mu }[s_r]⟩$. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions $\lambda$ with ${\lambda }_1\le 3$, we find a combinatorial interpretation for the multiplicities of the irreducible ${\mathfrak{S}}_n$-submodules of the Schur module ${\mathbb{S}}^{\lambda }{\mathbb{C}}^n$, considered as an ${\mathfrak{S}}_n$-module.