A generalized SXP rule proved by bijections and involutions

This paper proves a combinatorial rule expressing the product $s_{\tau }(s_{\lambda \mathrm{/}\mu }\circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $s_{\lambda }\circ p_r$. Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono, A simple proof of the Littlewood–Richardson rule and applications, Discrete Mathematics 193 (1998) 257–266. The connections with two later combinatorial rules for special cases of this plethysm are discussed. Two open problems are raised. The paper is intended to be readable by non-experts.