Parking functions and tree inversions revisited

TL;DR

This paper extends the combinatorial understanding of parking functions by relating their enumeration to tree inversions, generalizing previous bijective proofs to vector parking functions and clarifying their connection to graphical parking functions.

Contribution

It introduces a new expression for the reversed sum enumerator of vector parking functions using tree inversions, generalizing prior results and clarifying relationships between different parking function types.

Findings

Reversed sum enumerator for vector parking functions expressed via tree inversions.

Extended bijective proof to vector parking functions.

Clarified relationship between graphical and vector parking functions.

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.