Bilateral parking procedures

TL;DR

This paper introduces bilateral parking procedures on the integer line, exploring generalized parking rules that include parking to the left, and connects these procedures to combinatorial structures like labeled binary forests.

Contribution

It defines a new class of parking procedures allowing parking on both sides and establishes their combinatorial properties, including enumeration and encoding methods.

Findings

Number of parking functions equals (r+1)^{r-1} for certain local procedures.

Probabilistic bilateral procedures are analyzed.

Connections to labeled binary forests provide combinatorial insights.

Abstract

We introduce the class of bilateral parking procedures on the integer line. While cars try to park in the nearest available spot to their right in the classical case, we consider more general parking rules that allow cars to use the nearest available spot to their left. We show that for a natural subclass of local procedures, the number of corresponding parking functions of length $r$ is always equal to $(r+1)^{r-1}$. The setting can be extended to probabilistic procedures, in which the decision to park left or right is random. We finally describe how bilateral procedures can naturally be encoded by certain labeled binary forests, whose combinatorics shed light on several results from the literature.