Polyhedral geometry of refined q, t-Catalan numbers

We study a refinement of the q, t-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined q, t-Catalan numbers depend on a vector of parameters k and the classical q, t-Catalan numbers are recovered when k = (1, , 1). We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on k-Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on q, t-symmetry of the refined q, t-Catalan numbers in the cases where k = (k₁, k₂, k₃) and (k, k, k, k), give some extensions, including the case k = (k, k+m, k+m, k+m), and discuss relationships to other generalizations of the q, t-Catalan numbers.