Characters and chromatic symmetric functions (Q831348)

Summary: Let \(P\) be a poset, \( \text{inc}(P)\) its incomparability graph, and \(X_{\text{inc}(P)}\) the corresponding chromatic symmetric function, as defined by \textit{R. P. Stanley} in [Adv. Math. 111, No. 1, 166--194 (1995; Zbl 0831.05027)]. Let \(\omega\) be the standard involution on symmetric functions. We express coefficients of \(X_{\text{inc}(P)}\) and \(\omega X_{\text{inc}(P)}\) as character evaluations to obtain simple combinatorial interpretations of the power sum and monomial expansions of \(\omega X_{\text{inc}(P)}\) 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 \(X_{\text{inc}(P),q}\) when \(P\) is a unit interval order.