Symmetry of the refined $q,t$-Catalan polynomials for $\vec{k}$-Dyck paths

Abstract

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an involution on plane trees. They also provided a new formula for the original $q,t$-Catalan polynomials $C_{n}(q,t)$. We observe that depth is a slight modification of bounce, which was defined by the filling algorithm and ranking algorithm of Xin and the second author in their study of $\vec{k}$-Dyck paths. In this article, we generalize depth of classical Dyck paths to the case of $\vec{k}$-Dyck paths and prove $q,t$-symmetry of the pair of statistics (area, depth) for $\mathcal{K}$-Dyck paths. We provide an alternative description of the higher $q,t$-Catalan polynomials $C_{n}^{(k)}(q,t)$.