A group action on cyclic compositions and $γ$-positivity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909124909858816 |
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| author | Fu, Shishuo Yang, Jie |
| author_facet | Fu, Shishuo Yang, Jie |
| contents | Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, $2$-dominant compositions) is introduced, which amounts to give a combinatorial proof of the $γ$-positivity of the Narayana polynomial, as well as the $γ$-positivity of the polynomial $W_{2k+1,k}(t):=\sum_{1\le m\le k}w_{2k+1,k,m}t^m$ previously obtained by Bóna et al, with apparently new combinatorial interpretations of their $γ$-coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A group action on cyclic compositions and $γ$-positivity Fu, Shishuo Yang, Jie Combinatorics 05A05, 05A10, 05A15, 05C05 Let $w_{n,k,m}$ be the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$. We establish in two ways a new interpretation of the numbers $w_{n,k,m}$ in terms of plane trees and internal nodes. The first way builds on a new characterization of plane trees that involves cyclic compositions. The second proof utilizes a known interpretation of $w_{n,k,m}$ in terms of plane trees and leaves, and a recent involution on plane trees constructed by Li, Lin, and Zhao. Moreover, a group action on the set of cyclic compositions (or equivalently, $2$-dominant compositions) is introduced, which amounts to give a combinatorial proof of the $γ$-positivity of the Narayana polynomial, as well as the $γ$-positivity of the polynomial $W_{2k+1,k}(t):=\sum_{1\le m\le k}w_{2k+1,k,m}t^m$ previously obtained by Bóna et al, with apparently new combinatorial interpretations of their $γ$-coefficients. |
| title | A group action on cyclic compositions and $γ$-positivity |
| topic | Combinatorics 05A05, 05A10, 05A15, 05C05 |
| url | https://arxiv.org/abs/2403.00378 |