Mathematics – Combinatorics
Scientific paper
2008-03-03
Mathematics
Combinatorics
extended version
Scientific paper
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.
Cameron Peter J.
Johannsen Daniel
Prellberg Thomas
Schweitzer Pascal
No associations
LandOfFree
Counting Defective Parking Functions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Counting Defective Parking Functions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Counting Defective Parking Functions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-225823