On Weary Drivers, Records of Trees, and Parking Functions

TL;DR

This paper introduces a new perspective on parking functions via rooted trees, establishing a bijection that preserves records and demonstrating statistical equivalences, with some enumeration insights.

Contribution

It provides an alternative definition of parking functions, a record-preserving bijection with rooted trees, and a join equidistribution result for specific statistics.

Findings

Established a record-preserving bijection between rooted trees and parking functions.

Proved a join equidistribution between 5-tuples of statistics on both structures.

Explored enumerative questions related to parking functions and rooted trees.

Abstract

This work builds on the notion of record of rooted trees. We provide an alternative definition of parking functions, derive from it a record-preserving bijection between rooted trees and parking functions, and establish a join equidistribution result between a 5-tuple of statistics on rooted trees and a corresponding 5-tuple of statistics on parking functions. Some enumerative questions are also considered.