A group action on cyclic compositions and γ-positivity

Let wn,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 wn,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 wn,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 W2k+1,k(t):=∑1≤ m≤ kw2k+1,k,mtᵐ previously obtained by B\'{o}na et al, with apparently new combinatorial interpretations of their γ-coefficients.