Parking functions, multi-shuffle, and asymptotic phenomena

TL;DR

This paper introduces a new combinatorial construction called parking function multi-shuffle, characterizes multiple parking coordinates, and explores asymptotic properties of uniform $u$-parking functions under specific growth conditions.

Contribution

It presents a novel multi-shuffle construction for $u$-parking functions and analyzes their asymptotic probabilistic behavior in different regimes.

Findings

Explicit characterization of multiple parking coordinates.

Asymptotic properties differ sharply between $c>0$ and $c=0$ scenarios.

Provides combinatorial and probabilistic insights into parking functions.

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 $\lambda_i\leq u_i$ for all $1\leq i\leq m$. 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 $c=0$.