The e-positivity of the chromatic symmetric functions and the inverse Kostka matrix

This paper extends the current investigations on the famous long-time open conjectures by Stanley-Stembridge and Shareshian–Wachs with the q-parametrized version. We expand the chromatic symmetric functions for Dyck paths of bounce number three in the elementary symmetric function basis using a combinatorial interpretation of the inverse of the Kostka matrix studied in Egecioglu-Remmel (1990). We construct sign-reversing involutions to prove that certain coefficients in this expansion are positive. We use a similar method to establish the e-positivity of chromatic symmetric functions for Dyck paths of bounce number three beyond the "hook-shape" case of Cho-Huh (2019). Our results provide more supportive evidence for Stanley-Stembridge Conjecture by extending the e-positive class of the incomparability graph of natural unit interval orders.