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We expand the chromatic symmetric functions for Dyck paths of bounce numberthree in the elementary symmetric function basis using a combinatorialinterpretation of the inverse of the Kostka matrix studied inE\u{g}ecio\u{g}lu-Remmel (1990). We prove that certain coefficients in thisexpansion are positive. We establish the $e$-positivity of an extended class ofchromatic symmetric functions for Dyck paths of bounce number three beyond the"hook-shape" case of Cho-Huh (2019).

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