Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911451566833664 |
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| author | Siegl, Isaiah |
| author_facet | Siegl, Isaiah |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis Siegl, Isaiah Combinatorics 05E05 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. |
| title | Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2509.02841 |