On the expansion of the multiplicity-free plethysms $ p_{2}[s_{(a, b)} ] $ and $ p_{2}[s_{(1^{r}, 2^{t})}] $

Abstract

We show how to compute the explicit expansion of the plethysm $ p_{2}[s_{\lambda}]$ of the power symmetric function $ p_{2}$ and the Schur function $ s_{\lambda}$, where $\lambda$ has either two rows or two columns, via the well known Littlewood-Richardson coefficients which occur in the decomposition of $ s^{2}_{\lambda}$.