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.
\u02daReceived by the editors September 24, 2018; accepted for publication (in revised form) August 22, 2019; published electronically November 21, 2019. The extended abstract of this paper appeared in the proceedings of ``Formal Power Series and Algebraic Combinatorics 2018\\ (FPSAC'18) [1]. https://doi.org/10.1137/18M1216201 Funding: This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01057476). :Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea (chosj@ajou. ac.kr). ;Corresponding author. Applied Algebra and Optimization Research Center, Sungkyunkwan University, Suwon 16420, Republic of Korea (hyunyjia@g.skku.edu).
