Towards Upper and Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

Abstract: Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are $e$-positive. Finding a combinatorial interpretation for these $e$-coefficients remains a major open problem. Towards this goal, we introduce sets of strong and powerful $P$-tableaux and conjecture that these sets undercount and overcount the $e$-coefficients of the chromatic symmetric function $X_{inc(P)}(\mathbf{x}, q)$. We then use this framework to obtain combinatorial interpretations for various cases of $e$-expansion coefficients of chromatic symmetric functions of unit interval graphs. Additionally, we show that strong $P$-tableaux and the Shareshian--Wachs inversion statistic appear naturally in the proof of Hikita's result.