On e-positivity and e-unimodality of chromatic quasi-symmetric functions

Abstract

We use P-tableaux to give a combinatorial proof of the e-positivity of chromatic quasi-symmetric functions with bounce number two and some of those with bounce number three, which enables us to give a combinatorial model of the coefficients in the e-expansion of chromatic quasi-symmetric functions. We also find a combinatorial model for the e-coefficients of chromatic quasi-symmetric functions we considered, in terms of acyclic orientations of the corresponding graphs. For some special subclass of chromatic quasi-symmetric function s we considered for their e-positivity, we derive closed form formulae for the e-coefficients, showing the e-unimodality of the functions.