On statistics of prime parking functions, {\L}ukasiewicz paths, and quasisymmetric functions

TL;DR

This paper explores prime parking functions through combinatorial formulas, path correspondences, and algebraic links to quasisymmetric functions, revealing new structural insights and explicit average displacement calculations.

Contribution

It introduces explicit formulas for displacement and ties, establishes bijections with Lukasiewicz paths, and connects prime parking functions to Schur and quasisymmetric functions.

Findings

Derived an explicit average displacement formula for prime parking functions.

Established a bijection between parking functions and Lukasiewicz paths.

Linked Schur functions to fundamental quasisymmetric functions via prime parking function ties.

Abstract

We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over {\L}ukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeled{\L}ukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$