Symmetric decompositions and Euler-Stirling statistics on Stirling permutations
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866912500937654272 |
|---|---|
| author | Ma, Shi-Mei Wang, Jianfeng Yan, Guiying Yeh, Jean Yeh, Yeong-Nan |
| author_facet | Ma, Shi-Mei Wang, Jianfeng Yan, Guiying Yeh, Jean Yeh, Yeong-Nan |
| contents | The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(α,β)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian polynomials (J. Combin. Theory Ser. A, 2026), we present several symmetric decompositions of the enumerators related to Euler-Stirling statistics on Stirling permutations. Firstly, we provide a partial symmetric decomposition for the $1/k$-Eulerian polynomial. Secondly, we give several unexpected applications of the $(p,q)$-Eulerian polynomials, where $p$ marks the number of fixed points of permutations and $q$ marks that of cycles. From this paper, one can see that $(p,q)$-Eulerian polynomial contains a great deal of information about permutations and Stirling permutations. Using the change of grammars, we show that the $(α,β)$-Eulerian polynomials introduced by Carlitz-Scoville can be deduced from the $(p,q)$-Eulerian polynomials by special parametrizations. We then introduce proper and improper ascent-plateau statistics on Stirling permutations. Moreover, we introduce proper ascent, improper ascent, proper descent and improper descent statistics on permutations. Furthermore, we consider the joint distributions of Euler-Stirling statistics on permutations, including the numbers of improper ascents, proper ascents, left-to-right minima and right-to-left minina. In the final part, we first give a symmetric decomposition of the joint distribution of the ascent-plateau and left ascent-plateau statistics, and then we show that the $q$-ascent-plateau polynomials are bi-$γ$-positive, where $q$ marks the number of left-to-right minima. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17667 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetric decompositions and Euler-Stirling statistics on Stirling permutations Ma, Shi-Mei Wang, Jianfeng Yan, Guiying Yeh, Jean Yeh, Yeong-Nan Combinatorics 05A19, 05E05 The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(α,β)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian polynomials (J. Combin. Theory Ser. A, 2026), we present several symmetric decompositions of the enumerators related to Euler-Stirling statistics on Stirling permutations. Firstly, we provide a partial symmetric decomposition for the $1/k$-Eulerian polynomial. Secondly, we give several unexpected applications of the $(p,q)$-Eulerian polynomials, where $p$ marks the number of fixed points of permutations and $q$ marks that of cycles. From this paper, one can see that $(p,q)$-Eulerian polynomial contains a great deal of information about permutations and Stirling permutations. Using the change of grammars, we show that the $(α,β)$-Eulerian polynomials introduced by Carlitz-Scoville can be deduced from the $(p,q)$-Eulerian polynomials by special parametrizations. We then introduce proper and improper ascent-plateau statistics on Stirling permutations. Moreover, we introduce proper ascent, improper ascent, proper descent and improper descent statistics on permutations. Furthermore, we consider the joint distributions of Euler-Stirling statistics on permutations, including the numbers of improper ascents, proper ascents, left-to-right minima and right-to-left minina. In the final part, we first give a symmetric decomposition of the joint distribution of the ascent-plateau and left ascent-plateau statistics, and then we show that the $q$-ascent-plateau polynomials are bi-$γ$-positive, where $q$ marks the number of left-to-right minima. |
| title | Symmetric decompositions and Euler-Stirling statistics on Stirling permutations |
| topic | Combinatorics 05A19, 05E05 |
| url | https://arxiv.org/abs/2507.17667 |