On $e$-Positivity and $e$-Unimodality of Chromatic Quasi-symmetric Functions

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 functions we considered for their $e$-positivity, we derive closed form formulae for the $e$-coefficients, showing the $e$-unimodality of the functions.