Mahonian statistics on words with fixed weak right-to-left minima and on permutations with a fixed descent set
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914494306844672 |
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| author | Liu, Shao-Hua |
| author_facet | Liu, Shao-Hua |
| contents | Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed:
1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$, $\textsf{den}$, $\textsf{mak}$, $\textsf{mad}$, $\textsf{inv}_{r}$, $r\textsf{maj}$, and $r\textsf{den}$;
2. Euler--Mahonian statistics: $(\textsf{des},\textsf{maj})$, $(\textsf{exc},\textsf{den})$, and $(\textsf{des},\textsf{mak})$;
3. $r$-Euler--Mahonian statistics: $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$.
Our second main result shows that, for permutations with a fixed descent set, the statistics within each of the following two classes are equidistributed:
1. Mahonian statistics: $\textsf{inv}$, $\textsf{imaj}$, $\textsf{imak}$, $\textsf{iinv}_{r}$, and $\textsf{istat}$;
2. Mahonian--Stirling-type statistics: $(\textsf{inv},\textsf{lrmax})$, $(\textsf{imaj},\textsf{lrmax})$, $(\textsf{imak},\textsf{lrmax})$, and $(\textsf{iinv}_{r},\textsf{lrmax})$.
Moreover, we apply these results to set partitions, 221-avoiding words, alternating permutations, and permutations with $k$ alternating runs, thereby obtaining several families of equidistributed Mahonian-type statistics on these combinatorial structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18979 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mahonian statistics on words with fixed weak right-to-left minima and on permutations with a fixed descent set Liu, Shao-Hua Combinatorics 05A05, 05A19 Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$, $\textsf{den}$, $\textsf{mak}$, $\textsf{mad}$, $\textsf{inv}_{r}$, $r\textsf{maj}$, and $r\textsf{den}$; 2. Euler--Mahonian statistics: $(\textsf{des},\textsf{maj})$, $(\textsf{exc},\textsf{den})$, and $(\textsf{des},\textsf{mak})$; 3. $r$-Euler--Mahonian statistics: $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Our second main result shows that, for permutations with a fixed descent set, the statistics within each of the following two classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{imaj}$, $\textsf{imak}$, $\textsf{iinv}_{r}$, and $\textsf{istat}$; 2. Mahonian--Stirling-type statistics: $(\textsf{inv},\textsf{lrmax})$, $(\textsf{imaj},\textsf{lrmax})$, $(\textsf{imak},\textsf{lrmax})$, and $(\textsf{iinv}_{r},\textsf{lrmax})$. Moreover, we apply these results to set partitions, 221-avoiding words, alternating permutations, and permutations with $k$ alternating runs, thereby obtaining several families of equidistributed Mahonian-type statistics on these combinatorial structures. |
| title | Mahonian statistics on words with fixed weak right-to-left minima and on permutations with a fixed descent set |
| topic | Combinatorics 05A05, 05A19 |
| url | https://arxiv.org/abs/2604.18979 |