Probabilizing Parking Functions

TL;DR

This paper investigates the probabilistic structure of parking functions, offering new interpretations, combinatorial counts, and connections to Brownian excursions, enhancing understanding of their distributional properties.

Contribution

It introduces novel probabilistic interpretations of generating functions, counts parking functions with specific initial conditions, and links parking functions to Brownian excursion theory.

Findings

Derived new combinatorial counts for parking functions starting with a given number

Identified features with identical distributions among parking functions and all functions

Established links and differences between parking functions and Brownian excursions

Abstract

We explore the link between combinatorics and probability generated by the question "What does a random parking function look like?" This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It leads to new combinatorics: how many parking functions begin with $i$? We classify features (e.g., the full descent pattern) of parking functions that have exactly the same distribution among parking functions as among all functions. Finally, we develop the link between parking functions and Brownian excursion theory to give examples where the two ensembles differ.