Cycle structure of random parking functions

Abstract

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.

Parking Functions, Multi-shuffle, and Asymptotic Phenomena

“…Take (1,2), and α 3 = (5, 3) ∈ PF 2 (u 6 − u 5 , u 7 − u 5 ) = PF 2 (4,7). Then (11,13,5,1,6) ∈ MS(4, 8) is a multi-shuffle of the three words (1), (5,6), and (13,11).…”

“…Our asymptotic investigations in this work have concentrated mostly on the law of multiple coordinates and displacement of uniformly random (a, b)-parking functions, but there are many other structural properties of parking functions worth studying, for example, descent patterns, cycle counts, and equality processes, among others. The asymptotic distribution of some of these quantities have been explored in [2,5,13] for ordinary parking functions, but more generalized models are yet to be examined. Such models include u-parking functions studied in this paper, as well as parking functions on mappings, on graphs, and the effect of group action on parking functions.…”