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

TL;DR

This paper introduces a new family of skeletal paths and parking functions that generalize classical combinatorial objects, connecting them to chip firing, representation theory, and tropical geometry.

Contribution

It defines k-skeletal paths and parking functions, generalizing Dyck paths and classical parking functions, with bijections and connections to chip firing and advanced mathematical theories.

Findings

k-skeletal paths counted by Catalan numbers

k-skeletal parking functions match spanning tree counts

generalizations to non-integer parameters c and m

Abstract

For $0\leq k\leq n-1$, 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.