Restriction coefficients for partitions with at most three columns

Abstract

Let $r \geq 0$, and let $λ$ and $μ$ be partitions such that $λ_1 \leq r + 1$. We present a combinatorial interpretation of the plethysm coefficient $\langle s_λ, s_μ[s_r] \rangle$. As a consequence, we solve the restriction problem for partitions with at most three columns. That is, for all partitions $λ$ with $λ_1 \leq 3$, we find a combinatorial interpretation for the multiplicities of the irreducible $\mathfrak{S}_n$-submodules of the Schur module $\mathbb{S}^λ\mathbb{C}^n$, considered as an $\mathfrak{S}_n$-module.