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.