Bilateral parking procedures
Philippe Nadeau
TL;DR
This work expands classical parking functions by introducing bilateral parking procedures on $\mathbb{Z}$ and studying both deterministic and probabilistic rules. It identifies a universal enumeration: for any bilateral, local procedure, the number of parking functions of length $r$ is $$(r+1)^{r-1}$$, with a proof via a cyclic-argument construction that extends to probabilistic settings and abelian cases. A key contribution is the encoding of bilateral procedures by labeled binary forests through the $\widehat{\mathcal{P}}$ correspondence, which links parking dynamics to tree combinatorics and recovers known bijections in the classical case. The paper also develops colored extensions and demonstrates how the framework encompasses various known and new rules (e.g., Naples variants) while connecting to remixed Eulerian numbers. Altogether, it provides a unifying, structurally rich approach to bilateral (and local) parking, with broad combinatorial implications and natural extensions to probabilistic and colored settings.
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.