The spiders $S(4m+2,\,2m,\,1)$ are $e$-positive

TL;DR

This paper proves the $e$-positivity of a specific class of spiders $S(4m+2, 2m, 1)$ using a composition method and an explicit injection, confirming a conjecture in algebraic combinatorics.

Contribution

It introduces a novel composition-based approach and explicit injection construction to establish $e$-positivity for the specified spider graphs, advancing the understanding of symmetric functions.

Findings

Confirmed $e$-positivity of $S(4m+2, 2m, 1)$ spiders

Developed a composition and divide-and-conquer method

Constructed explicit injections for positivity proof

Abstract

By using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, van Willigenburg and Wang. Following the divide-and-conquer strategy, we group one or two $e_J$-terms that have positive coefficients with each $e_I$-term that has a negative coefficient, where the compositions $J$ are selected to be obtained by rearranging the parts of $I$, and show the positivity of the sum of those coefficients. Our main contribution is an explicit construction of the injection.