A Combinatorial Interpretation for the coefficients in the Kronecker Product $s_{(n-p,p)}\ast s_{\lambda }$ (Multiplicities in the Kronecker Product $s_{(n-p,p)}\ast s_{\lambda }$)

In this paper we give a combinatorial interpretation for the coefficient of $s_{\nu }$ in the Kronecker product $s_{(n-p,p)}\ast s_{\lambda }$, where $\lambda =({\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_{\mathrm{\ell }(\lambda )})⊢n$, if $\mathrm{\ell }(\lambda )\ge 2p-1$ or ${\lambda }_1\ge 2p-1$; that is, if $\lambda$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $\lambda$ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all $\lambda$ when $n>(2p-2{)}^2$. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker products. We have also obtained some formulas for special cases.