A group action on cyclic compositions and ${\hat{I}}^3$-positivity

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 ${\hat{I}}^3$-positivity of the Narayana polynomial, as well as the ${\hat{I}}^3$-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 ${\hat{I}}^3$-coefficients.

19 pages, 3 figures