A generalized SXP rule proved by bijections and involutions

TL;DR

This paper establishes a generalized combinatorial rule for a specific product of symmetric functions using bijections and involutions, extending previous results and connecting to other rules in algebraic combinatorics.

Contribution

It introduces a new generalized SXP rule for plethysm of skew Schur functions, proved via combinatorial bijections and involutions, expanding understanding of symmetric function products.

Findings

Proves a generalized SXP rule for plethysm involving skew Schur functions.

Uses explicit bijections and sign-reversing involutions in the proof.

Discusses connections with other combinatorial rules and open problems.

Abstract

This paper proves a combinatorial rule expressing the product $s_\tau(s_{\lambda/\mu} \circ p_r)$ 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.