Toward 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 -positive. Finding a combinatorial interpretation for these -coefficients remains a major open problem. One approach is to look for combinatorial interpretations which are subsets of Gasharov's -tableaux. Towards this goal, we introduce sets of strong and powerful -tableaux, and use them to find combinatorial interpretations for various -coefficients of the chromatic symmetric function . We conjecture that the set of strong -tableaux gives a lower bound for the -coefficients of . Additionally, we show that strong -tableaux and the Shareshian--Wachs inversion statistic appear naturally in the proof of Hikita's result.
Keywords
Cite
@article{arxiv.2509.02841,
title = {Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis},
author = {Isaiah Siegl},
journal= {arXiv preprint arXiv:2509.02841},
year = {2026}
}
Comments
In a previous version, we conjectured that powerful $P$-tableaux give an upper bound for the $e$-coefficients of chromatic symmetric functions of incomparability graphs of natural unit interval orders. An anonymous reviewer showed that the poset corresponding to the reverse Hessenberg function (0,0,1,1,2,3,4,6) gives a counterexample to this conjecture