On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations
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| Format: | Preprint |
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2024
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| _version_ | 1866914905660063744 |
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| author | Huang, Kaimei Lin, Zhicong Yan, Sherry H. F. |
| author_facet | Huang, Kaimei Lin, Zhicong Yan, Sherry H. F. |
| contents | A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and
$( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $(\mathrm{exc}_r, \mathrm{den}_r)$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over $\mathfrak{S}_{n}$, thereby confirming a recent conjecture posed by Liu. When $r=1$, the result recovers the equidistribution of $(\mathrm{des}, \mathrm{maj})$ and $(\mathrm{exc}, \mathrm{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04185 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations Huang, Kaimei Lin, Zhicong Yan, Sherry H. F. Combinatorics A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set $\mathfrak{S}_{n}$ of all permutations of $\{1,2,\ldots, n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $(\mathrm{exc}_r, \mathrm{den}_r)$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over $\mathfrak{S}_{n}$, thereby confirming a recent conjecture posed by Liu. When $r=1$, the result recovers the equidistribution of $(\mathrm{des}, \mathrm{maj})$ and $(\mathrm{exc}, \mathrm{den})$, which was first conjectured by Denert and proved by Foata and Zeilberger. |
| title | On a conjecture concerning the $r$-Euler-Mahonian statistic on permutations |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.04185 |