Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

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. One approach is to look for combinatorial interpretations which are subsets of Gasharov’s $P$-tableaux. Towards this goal, we introduce sets of strong and powerful $P$-tableaux, and use them to find combinatorial interpretations for various $e$-coefficients of the chromatic symmetric function $X_{inc(P)}(\mathbf{x},q)$. We conjecture that the set of strong $P$-tableaux gives a lower bound for the $e$-coefficients of $X_{inc(P)}(\mathbf{x},q)$. Additionally, we show that strong $P$-tableaux and the Shareshian–Wachs inversion statistic appear naturally in the proof of Hikita’s result.