Counting Defective Parking Functions

TL;DR

This paper analyzes defective parking functions, establishing recurrence relations, explicit formulas, and asymptotic behaviors, including a limiting Rayleigh distribution for the defect when choices are random and the number of spaces equals drivers.

Contribution

It derives a recurrence relation and explicit formulas for defective parking functions, and characterizes their asymptotic distribution, including the limiting Rayleigh distribution.

Findings

Explicit formula for the number of defective parking functions.

Asymptotic distribution of the defect is Rayleigh when m=n.

Probability all spaces are occupied approaches one when m grows faster than n.

Abstract

Suppose that $n$ drivers each choose a preferred parking space in a linear car park with $m$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a \emph{defective parking function} of \emph{defect} $k$. Let $\cp(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers $\cp(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of $\cp(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in case $m=\omega(n)$, the probability that all spaces are occupied tends asymptotically to one.