Increasing Binary Trees and the $(α,β)$-Eulerian Polynomials
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912298089578496 |
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| author | Chen, William Y. C. Fu, Amy M. |
| author_facet | Chen, William Y. C. Fu, Amy M. |
| contents | In light of the grammar given by Ji for the $(α,β)$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the $γ$-coefficients of the $α$-Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the $γ$-coefficients of the derangement polynomials in the same vein. By means of a decomposition of an increasing binary tree into a forest, we find combinatorial interpretations of the sums involving two identities of Ji, one of which can be viewed as $(α,β)$-extensions of the formulas of Petersen and Stembridge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10331 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Increasing Binary Trees and the $(α,β)$-Eulerian Polynomials Chen, William Y. C. Fu, Amy M. Combinatorics 05A15, 05A19 In light of the grammar given by Ji for the $(α,β)$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the $γ$-coefficients of the $α$-Eulerian polynomials in terms of forests of planted 0-1-2-plane trees, which specializes to a combinatorial interpretation of the $γ$-coefficients of the derangement polynomials in the same vein. By means of a decomposition of an increasing binary tree into a forest, we find combinatorial interpretations of the sums involving two identities of Ji, one of which can be viewed as $(α,β)$-extensions of the formulas of Petersen and Stembridge. |
| title | Increasing Binary Trees and the $(α,β)$-Eulerian Polynomials |
| topic | Combinatorics 05A15, 05A19 |
| url | https://arxiv.org/abs/2404.10331 |