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A group action on cyclic compositions and $\gamma$-positivity
- Publication Year :
- 2024
-
Abstract
- 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 $\gamma$-positivity of the Narayana polynomial, as well as the $\gamma$-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 B\'{o}na et al, with apparently new combinatorial interpretations of their $\gamma$-coefficients.<br />Comment: 19 pages, 3 figures
- Subjects :
- Mathematics - Combinatorics
05A05, 05A10, 05A15, 05C05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2403.00378
- Document Type :
- Working Paper