Distribution of new statistics of parking functions and their generalizations

TL;DR

This paper introduces new statistics on parking functions and labeled forests, providing their joint distributions, explaining recent equidistribution phenomena, and establishing bijections, thereby advancing combinatorial enumeration methods.

Contribution

It presents novel statistics on parking functions, extends them to labeled forests, and explains recent equidistribution results through explicit bijections.

Findings

Joint distribution of two new statistics on parking functions

Explicit bijection between two related statistics

Extension of techniques to further refined statistics

Abstract

In this paper we present new results on the enumeration of parking functions and labeled forests. We introduce new statistics on parking functions, which are then extended to labeled forests via bijective correspondences. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics. Extensions of our techniques are discussed, including joint distribution on further refinement of these new statistics.