Further results on $r$-Euler-Mahonian statistics
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arXiv
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| Natura: | Preprint |
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2025
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| author | Huang, Kaimei Yan, Sherry H. F. |
| author_facet | Huang, Kaimei Yan, Sherry H. F. |
| contents | As natural generalizations of the descent number ($\des$) and the major index ($\maj$), Rawlings introduced the notions of the $r$-descent number ($r\des$) and the $r$-major index ($r\maj$) for a given positive integer $r$. A pair $(\st_1, \st_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $ (\mathrm{st_1}, \mathrm{st_2})$ and
$ (r\des, r\maj)$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that $(g\exc_\ell, g\den_\ell)$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$, where $g\exc_\ell$ denotes the $g$-gap $\ell$-level excedance number and $g\den_\ell$ denotes the $g$-gap $\ell$-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of $(g\exc_\ell, g\den_\ell)$ and $ (r\des, r\maj)$ where $r=g+\ell-1$.
Setting $g=\ell=1$, our result recovers the equidistribution of $(\des, \maj)$ and $(\exc, \den)$, which was first conjectured by Denert and proved by Foata and Zeilberger. Our second main result is concerned with the analogous result for $(g\exc_\ell, g\den_{g+\ell})$ which states that $(g\exc_\ell, g\den_{g+\ell})$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12083 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Further results on $r$-Euler-Mahonian statistics Huang, Kaimei Yan, Sherry H. F. Combinatorics As natural generalizations of the descent number ($\des$) and the major index ($\maj$), Rawlings introduced the notions of the $r$-descent number ($r\des$) and the $r$-major index ($r\maj$) for a given positive integer $r$. A pair $(\st_1, \st_2)$ of permutation statistics is said to be $r$-Euler-Mahonian if $ (\mathrm{st_1}, \mathrm{st_2})$ and $ (r\des, r\maj)$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that $(g\exc_\ell, g\den_\ell)$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$, where $g\exc_\ell$ denotes the $g$-gap $\ell$-level excedance number and $g\den_\ell$ denotes the $g$-gap $\ell$-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of $(g\exc_\ell, g\den_\ell)$ and $ (r\des, r\maj)$ where $r=g+\ell-1$. Setting $g=\ell=1$, our result recovers the equidistribution of $(\des, \maj)$ and $(\exc, \den)$, which was first conjectured by Denert and proved by Foata and Zeilberger. Our second main result is concerned with the analogous result for $(g\exc_\ell, g\den_{g+\ell})$ which states that $(g\exc_\ell, g\den_{g+\ell})$ is $(g+\ell-1)$-Euler-Mahonian for all positive integers $g$ and $\ell$. |
| title | Further results on $r$-Euler-Mahonian statistics |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.12083 |