On the $\operatorname{rix}$ statistic and valley-hopping
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916147097501696 |
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| author | Lafrenière, Nadia Zhuang, Yan |
| author_facet | Lafrenière, Nadia Zhuang, Yan |
| contents | This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $γ$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $Φ$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02711 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $\operatorname{rix}$ statistic and valley-hopping Lafrenière, Nadia Zhuang, Yan Combinatorics 05A05 (Primary), 05E18 (Secondary) This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $γ$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $Φ$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping. |
| title | On the $\operatorname{rix}$ statistic and valley-hopping |
| topic | Combinatorics 05A05 (Primary), 05E18 (Secondary) |
| url | https://arxiv.org/abs/2307.02711 |