On e-positivity of the chromatic symmetric function

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.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/