Lucky cars and lucky spots in parking functions

TL;DR

This paper investigates the probability and asymptotic behavior of lucky cars and spots in parking functions, extending previous work on lucky cars and exploring new complexities of lucky spots, especially in ordered preference scenarios.

Contribution

It introduces a detailed analysis of lucky spots in parking functions, expanding understanding beyond lucky cars, and provides probabilistic results for specific positions and preference orderings.

Findings

Probabilities for lucky spots at initial positions are derived.

Asymptotic behavior of lucky spots is characterized.

Special cases for ordered preferences are analyzed.

Abstract

Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. When a car parks in its preferred spot then the corresponding car and corresponding spot are deemed ``lucky.'' This paper looks briefly at lucky cars which have previously been studied and in simple cases can be understood by a generalization of a result due to Pollak. We also consider lucky spots where the situation is more complex and not previously studied. Probabilities and asymptotics for lucky spots are given for the first few spots on the one-way street. We close with an exploration of the special cases when cars enter the one-way street in either weakly-increasing or weakly-decreasing order of their preferences.