Counting pattern-avoiding permutations by big descents
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913486124089344 |
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| author | Elizalde, Sergi Rivera Jr., Johnny Zhuang, Yan |
| author_facet | Elizalde, Sergi Rivera Jr., Johnny Zhuang, Yan |
| contents | A descent $k$ of a permutation $π=π_{1}π_{2}\dotsπ_{n}$ is called a big descent if $π_{k}>π_{k+1}+1$; denote the number of big descents of $π$ by $\operatorname{bdes}(π)$. We study the distribution of the $\operatorname{bdes}$ statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets $Π\subseteq\mathfrak{S}_{3}$ of size 1 and 2 into $\operatorname{bdes}$-Wilf equivalence classes, and we derive a formula for the distribution of big descents for each of these classes. Our methods include generating function techniques along with various bijections involving objects such as Dyck paths and binary words. Several future directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15111 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting pattern-avoiding permutations by big descents Elizalde, Sergi Rivera Jr., Johnny Zhuang, Yan Combinatorics Primary 05A15, Secondary 05A05, 05A19 A descent $k$ of a permutation $π=π_{1}π_{2}\dotsπ_{n}$ is called a big descent if $π_{k}>π_{k+1}+1$; denote the number of big descents of $π$ by $\operatorname{bdes}(π)$. We study the distribution of the $\operatorname{bdes}$ statistic over permutations avoiding prescribed sets of length-three patterns. Specifically, we classify all pattern sets $Π\subseteq\mathfrak{S}_{3}$ of size 1 and 2 into $\operatorname{bdes}$-Wilf equivalence classes, and we derive a formula for the distribution of big descents for each of these classes. Our methods include generating function techniques along with various bijections involving objects such as Dyck paths and binary words. Several future directions of research are proposed, including conjectures concerning real-rootedness, log-concavity, and Schur positivity. |
| title | Counting pattern-avoiding permutations by big descents |
| topic | Combinatorics Primary 05A15, Secondary 05A05, 05A19 |
| url | https://arxiv.org/abs/2408.15111 |