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

Abstract

Using the composition method, we establish the $e$-positivity of spiders of the form $S(4m+2,2m,1)$, as conjectured by Aliniaeifard, van Willigenburg and Wang. Following a divide-and-conquer strategy, we group one or two positive-coefficient $e_J$-terms with each negative-coefficient $e_I$-term, where each $J$ is a composition obtained by rearranging the parts of the composition $I$, and prove that the resulting total coefficient is nonnegative. Our main contribution is the explicit construction of the required injections.

arXiv comment22 pages, 4 figures