On the largest Littlewood--Richardson coefficient

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