Kronecker Coefficients For Some Near-Rectangular Partitions

Abstract

We give formulae for computing Kronecker coefficients occurring in the expansion of $s_{\mu }\ast s_{\nu }$, where both $\mu$ and $\nu$ are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study $s_{(n,n-1,1)}\ast s_{(n,n)}$, $s_{(n-1,n-1,1)}\ast s_{(n,n-1)}$, $s_{(n-1,n-1,2)}\ast s_{(n,n)}$, $s_{(n-1,n-1,1,1)}\ast s_{(n,n)}$ and $s_{(n,n,1)}\ast s_{(n,n,1)}$. Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers.