Chromatic quasisymmetric functions for signed graphs

Abstract

In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this function for a directed graph, that appears to be in $QSym$, the algebra of quasisymmetric functions, which is of great interest in algebraic combinatorics. Our goal is to extend this work to signed graphs, taking into account the perspective of the hyperplane arrangement associated with a signed graph, developed by Zaslavsky. We introduce the signed chromatic quasisymmetric invariant, and obtain structural properties. As a consequence, we define and study $SQSym$, the algebra of signed quasisymmetric functions.