UW Combinatorics and Geometry Seminar: Foster Tom

We prove a new signed elementary symmetric function expansion of the chromatic symmetric function. We then use sign-reversing involutions to prove $e$-positivity for graphs formed by joining cycles or cliques at single vertices. By considering connected partitions, we prove non-$e$-positivity for all trees that have either a vertex of degree at least five, or a vertex of degree four that is not adjacent to a leaf. We also prove that spiders with four legs are not $e$-positive.