Mathematics > Combinatorics
arXiv:2509.02841 (math)
[Submitted on 2 Sep 2025 (v1), last revised 31 Jul 2026 (this version, v3)]
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. 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.
| 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 |
| Subjects: | Combinatorics (math.CO) |
| MSC classes: | 05E05 |
| Cite as: | arXiv:2509.02841 [math.CO] |
| (or arXiv:2509.02841v3 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2509.02841 arXiv-issued DOI via DataCite |
Submission history
From: Isaiah Siegl [view email]
[v1]
Tue, 2 Sep 2025 21:20:53 UTC (38 KB)
[v2]
Tue, 17 Feb 2026 06:25:09 UTC (111 KB)
[v3]
Fri, 31 Jul 2026 22:16:29 UTC (119 KB)
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