Skeletal generalizations of Dyck paths, parking functions, and chip-firing games

For 0 k n-1, we introduce a family of k-skeletal paths which are counted by the n-th Catalan number for each k, and specialize to Dyck paths when k=n-1. We similarly introduce k-skeletal parking functions which are equinumerous with the spanning trees on n+1 vertices for each k, and specialize to classical parking functions for k=n-1. The preceding constructions are generalized to paths lying in a trapezoid with base c > 0 and southeastern diagonal of slope 1/m; c and m need not be integers. We give bijections among these families when k varies with m and c fixed. Our constructions are motivated by chip firing and have connections to combinatorial representation theory and tropical geometry.