A $q$-analog of the Stirling-Eulerian Polynomials
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913315521822720 |
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| author | Dong, Yao Lin, Zhicong Pan, Qiongqiong |
| author_facet | Dong, Yao Lin, Zhicong Pan, Qiongqiong |
| contents | In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|α,β)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|α,β)$, denoted $P_n(u_1,u_2,u_3,u_4|α,β)$, which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen's context-free grammar calculus, Ji proved a formula for the generating function of $P_n(u_1,u_2,u_3,u_4|α,β)$, generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing $γ$-positivity expansion for $A_n(x,y|α,β)$. In this paper, we prove a $q$-analog of Ji's formula by using Gessel's $q$-compositional formula and provide a combinatorial approach to her $γ$-positivity expansion of $A_n(x,y|α,β)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $q$-analog of the Stirling-Eulerian Polynomials Dong, Yao Lin, Zhicong Pan, Qiongqiong Combinatorics In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial $A_n(x,y|α,β)$ as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of $A_n(x,y|α,β)$, denoted $P_n(u_1,u_2,u_3,u_4|α,β)$, which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen's context-free grammar calculus, Ji proved a formula for the generating function of $P_n(u_1,u_2,u_3,u_4|α,β)$, generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing $γ$-positivity expansion for $A_n(x,y|α,β)$. In this paper, we prove a $q$-analog of Ji's formula by using Gessel's $q$-compositional formula and provide a combinatorial approach to her $γ$-positivity expansion of $A_n(x,y|α,β)$. |
| title | A $q$-analog of the Stirling-Eulerian Polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.09777 |