Counting Defective Parking Functions

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dc.creatorCameron, Peter J.
dc.creatorJohannsen, Daniel
dc.creatorPrellberg, Thomas
dc.creatorSchweitzer, Pascal
dc.date2008-03-03
dc.date2008-07-08
dc.date.accessioned2026-07-07T09:48:32Z
dc.date.available2026-07-07T09:48:32Z
dc.descriptionSuppose 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 $m=ω(n)$, the probability that all spaces are occupied tends asymptotically to one.
dc.descriptionextended version
dc.identifierhttps://arxiv.org/abs/0803.0302
dc.identifierhttp://arxiv.org/abs/0803.0302
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/164247
dc.subjectCombinatorics
dc.subject05A15, 05A16
dc.titleCounting Defective Parking Functions
dc.typetext

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