Parking functions, multi-shuffle, and asymptotic phenomena

Abstract

Given a positive-integer-valued vector $u=(u_1,\dots ,u_m)$ with $u_1<\cdots <u_m$. A $u$-parking function of length $m$ is a sequence $\pi =({\pi }_1,\dots ,{\pi }_m)$ of positive integers whose non-decreasing rearrangement $({\lambda }_1,\dots ,{\lambda }_m)$ satisfies ${\lambda }_i\le u_i$ for all $1\le i\le m$. 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 $u_i=cm+ib$. The asymptotic scenario in the generic situation $c>0$ is in sharp contrast with that of the special situation $c=0$.

arXiv comment19 pages, 2 figures. This article extends the multi-shuffle mechanism introduced in arXiv:2107.01767 to generic u-parking functions