Characters and Chromatic Symmetric Functions

Abstract

Let $P$ be a poset, $\mathrm{inc}(P)$ its incomparability graph, and $X_{\mathrm{inc}(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in Adv. Math., 111 (1995) pp.166–194. Let $\omega$ be the standard involution on symmetric functions.  We express coefficients of $X_{\mathrm{inc}(P)}$ and $\omega X_{\mathrm{inc}(P)}$ as character evaluations to obtain simple combinatorial interpretations of the power sum and monomial expansions of $\omega X_{\mathrm{inc}(P)}$ which hold for all posets $P$. Consequences include new combinatorial interpretations of the permanent, induced trivial character immanants, and power sum immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{\mathrm{inc}(P),q}$ when $P$ is a unit interval order.