2008/03/03 by Peter J. Cameron, Peter J. Cameron, Daniel Johannsen +6 · 3 citations
Computer Science · Mathematics · #05A15 #05A16 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Mathematics #math.CO #msc:05A15 #msc:05A16
paper · pdf · doi:10.48550/arxiv.0803.0302
extended version
openalex publication_date 2008/03/03 · arxiv created 2008/07/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 defective parking function of 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=ω(n), the probability that all spaces are occupied tends asymptotically to one.