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 $C_{n}^{(k)}(q,t)$.