On the largest Littlewood--Richardson coefficient

We study partitions which attain the largest Littlewood-Richardson coefficient. More precisely, we prove that the largest $c^\lambda_{\mu\nu}$ is attained at partitions such that $\mu \subseteq \nu$ and $\nu/\mu$ is a disjoint union of squares. We conjecture that for $n \ge 16$, all largest $c^\lambda_{\mu\nu}$ must satisfy this property. We confirm this conjecture numerically, for $16 \le n \le 45$.