Substring compatibility of permutation statistics
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909876326760448 |
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| author | Tang, Michael |
| author_facet | Tang, Michael |
| contents | A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Substring compatibility of permutation statistics Tang, Michael Combinatorics 05A05, 05E05, 16T30 A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture. |
| title | Substring compatibility of permutation statistics |
| topic | Combinatorics 05A05, 05E05, 16T30 |
| url | https://arxiv.org/abs/2510.25524 |