New $r$-Euler--Mahonian statistics involving Denert's statistic
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866909740425019392 |
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| author | Liu, Shao-Hua |
| author_facet | Liu, Shao-Hua |
| contents | Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Any pair of permutation statistics that is equidistributed with these pairs is said to be $r$-Euler--Mahonian. Several classes of $r$-Euler--Mahonian statistics were established by Huang--Lin--Yan and Huang--Yan. Inspired by their bijections, we provide a new bijective proof of the classical result that $(\textsf{exc},\textsf{den})$ is Euler--Mahonian. Using this bijection, we further show that $(\textsf{exc}_{r},\textsf{den})$ is $r$-Euler--Mahonian, where $\textsf{exc}_{r}$ denotes the number of $r$-level excedances (i.e., excedances at least $r$). Furthermore, by extending our bijection, we establish a more general result that encompasses all the aforementioned results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New $r$-Euler--Mahonian statistics involving Denert's statistic Liu, Shao-Hua Combinatorics Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Any pair of permutation statistics that is equidistributed with these pairs is said to be $r$-Euler--Mahonian. Several classes of $r$-Euler--Mahonian statistics were established by Huang--Lin--Yan and Huang--Yan. Inspired by their bijections, we provide a new bijective proof of the classical result that $(\textsf{exc},\textsf{den})$ is Euler--Mahonian. Using this bijection, we further show that $(\textsf{exc}_{r},\textsf{den})$ is $r$-Euler--Mahonian, where $\textsf{exc}_{r}$ denotes the number of $r$-level excedances (i.e., excedances at least $r$). Furthermore, by extending our bijection, we establish a more general result that encompasses all the aforementioned results. |
| title | New $r$-Euler--Mahonian statistics involving Denert's statistic |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.12717 |