Parking functions, multi-shuffle, and asymptotic phenomena

Abstract

Given a positive-integer-valued vector u = (u1, . . . , um) with u1 < • • • < um. A u-parking function of length m is a sequence π = (π1, . . . , πm) of positive integers whose nondecreasing rearrangement (λ1, . . . , λm) satisfies λi ≤ ui for all 1 ≤ i ≤ m. We introduce a combinatorial construction termed a parking function multi-shuffle to generic u-parking functions and obtain an explicit characterization of multiple parking coordinates. As an application, we derive various asymptotic probabilistic properties of a uniform u-parking function when ui = cm + ib. The asymptotic scenario in the generic situation c > 0 is in sharp contrast with that of the special situation c = 0.

Cycle structure of random parking functions

“…5.3. In [33] and [34], Yin initiated the probabilistic study of (m, n)-parking functions and u-parking functions, respectively, and in particular, obtained explicit formulas for their parking completions. It should be tractable to follow our approach and use Stein's method via exchangeable pairs to obtain limit theorems, with convergence rates, for the distribution of cycle counts in these generalized models.…”

“…In [20], Kenyon and Yin explored links between combinatorial and probabilistic aspects of parking functions. Extending previous work, Yin developed the parking function multi-shuffle to obtain formulas for parking completions, moments of multiple coordinates, and all possible covariances between two coordinates for (m, n)-parking functions (where there are m ≤ n cars and n spots) [33] and u-parking functions [34].…”