Combinatorial aspects of a discrete parking process

Abstract

This article examines a discrete parking process on a line of $n$ sites, where particles arrive sequentially, choosing a position uniformly at random among the currently available sites, and occupy a site only if it and its neighbors are vacant. The process ends when no further placements are possible, yielding a jamming limit configuration. We analyze combinatorial properties of the space of such configurations, including the enumeration of specific subsets. To contrast with the sequential dynamics, we introduce a static model in which all admissible jamming configurations are equally likely. For this model, we establish results on occupancy probabilities, the distribution and mean of occupied sites, and asymptotic occupancy density. A key finding is that the static model is stochastically dominated by the sequential one in terms of total occupancy. The study highlights structural and probabilistic differences between the models, contributing to the understanding of parking processes and random sequential adsorption.