Parking functions and tree inversions revisited

Abstract: Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that moreover generalizes to graphical parking functions. Using a depth-first search variant of Dhar's burning algorithm they proved that the codegree enumerator for $G$-parking functions equals the $\kappa$-number enumerator for spanning trees of $G$. The $\kappa$-number is a kind of generalized tree inversion number originally defined by Gessel. We extend the work of Perkinson-Yang-Yu to what are referred to as "generalized parking functions" in the literature, but which we prefer to call vector parking functions because they depend on a choice of vector $\mathbf{x} \in \mathbb{N}^n$. Specifically, we give an expression for the reversed sum enumerator for $\mathbf{x}$-parking functions in terms of inversions in rooted plane trees with respect to certain admissible vertex orders. Along the way we clarify the relationship between graphical and vector parking functions.