Probabilistic Parking Functions

TL;DR

This paper introduces a probabilistic extension of classical parking functions, analyzing their properties, phase transitions, and combinatorial implications, including solving an open problem and connecting to broader combinatorial phenomena.

Contribution

It extends parking functions to a probabilistic setting, proves independence of certain probabilities from parameters, and provides new combinatorial interpretations and solutions to open problems.

Findings

Probability of a preference vector being a parking function is independent of p.

Demonstrates a sharp transition in parking statistics at p=1/2.

Provides a combinatorial interpretation for OEIS A220884, solving an open problem.

Abstract

We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodr\'iguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots, which solves an open problem of Novelli and Thibon posed in 2020 (arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.