Counting Defective Parking Functions

Abstract

Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a 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 the case m = ω(n), the probability that all spaces are occupied tends asymptotically to one.

Parking function varieties for combinatorial tree models

“…Another promising direction is to study defective tree parking functions, i.e., to study the "overflow" of the number of drivers, which could not park successfully (see [3] for a respective treatment on ordinary parking functions). With the methods proposed it seems that the results presented could be extended to this problem and the author plans to comment on that in a future work.…”

“…w) • e nh(w) dw, withg(w) = (1 − 3w)( n−m 2n−1 + w) w(1 − w)(1 + w)3 , h(w) choose as contour in the contour integral representation a suitable simple positively oriented closed curve around the origin. To locate the saddle points in the relevant part e nh(w) of the integrand, we have to solve the equationh (w) = (3w − 1)(m − nw) nw(1 − w)(1 + w) = 0,yielding the solutions w 1 = m n and w 2 = 1 3…”

Partial parking functions

“…Konheim and Weiss [7] introduced the concept of parking functions that can be thus described. Suppose that n drivers want to park in a one-way street with exactly n places and that a ∈ [n] n is the record of the preferred parking slots, that is, a i is the preferred (3) With this definition, 1 := (1, . .…”

“…The restriction to [k, n] of a function a that parks all the n + 1 − k elements of [k, n] is a particular case of a defective parking function introduced by Cameron, Johannsen, Prellberg and Schweitzer [3]. Hence, the number T k of all functions that park every element of [k, n] is n k−1 c(n, n + 1 − k, 0), where c(n, m, k) is the number of (n, m, k)-defective parking functions [3, pp.3], that is T k = k n k−1 (n + 1) n−k .…”

Parking Functions on Directed Graphs and Some Directed Trees

“…It is well-known that there are (n+1) n−1 parking functions of length n, while there are (n−m+1)(n+1) m−1 classical (n, m)-parking functions [3]. Classical parking functions have appeared throughout combinatorics as chains in the noncrossing lattice, in enumeration of hyperplane arrangements, in noncrossing partitions, and in tree enumeration (see [5,12,14] as well as references herein).…”