On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions

TL;DR

The paper studies prime parking functions of length $n+1$ through multiple lenses, deriving exact and asymptotic displacement statistics and presenting a displacement-enumerator in terms of weighted Łukasiewicz paths. It establishes a one-to-one correspondence between (prime) parking functions and labeled Łukasiewicz and Dyck paths, enabling a detailed analysis of descents, ascents, and ties via $\ell$-forward differences, with key results showing $\mathbb{E}[\pi_1|\pi\in\mathrm{PPF}_{n+1}] = \frac{1}{2}\left(n+3-\frac{n!}{n^n}\sum_{s=0}^n\frac{n^{s}}{s!}\right)$ and $\mathbb{E}[\mathrm{dis}(\pi)|\pi\in\mathrm{PPF}_{n+1}] = \frac{\sqrt{2\pi}}{4}n^{3/2}-\frac{n}{6}+o(n)$. The displacement-enumerator for prime parking functions is given by a sum over prime Łukasiewicz paths, and a labeled-Lukasiewicz path framework yields direct descriptions of descent/ascents/ties. The authors further connect tie sets to quasisymmetric and Schur functions, showing $\sum_{\pi\in\mathrm{PPF}_n} F_{n,\mathrm{Tie}(\pi)} = \sum_{i=1}^n (n-2)^{i-1} s_{(i,1^{n-i})}$ and providing joint generating-function identities with $\Delta_{\ell}$ and $\Delta_{m}$. Overall, the work links prime parking functions to Łukasiewicz-path combinatorics and to (quasi)symmetric function theory, with potential implications for data-hashing models and representation-theoretic interpretations in algebraic combinatorics.

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 Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz 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 $n-i.$

Figures (4)

• Figure 1: The (prime) Ł ukasiewicz path corresponding to $\pi = (1,1,1,3,4,4,6) \in \mathop{\mathrm{PPF}}\nolimits_7$.
• Figure 2: The labeled Ł ukasiewicz path $L=(\ell,\beta)$ described in \ref{['ex:Luka to pf']}.
• Figure 3: A labeled Dyck path of length $8$ and its corresponding parking function.
• Figure 4: Illustration of the bijection in \ref{['thm:labeled luka bijection to PFs']}.

Theorems & Definitions (60)

• Theorem 1.1
• Corollary 1.1
• Corollary 1.1
• Theorem 1.2
• Proposition 1.2
• Theorem 1.3
• Example 1
• Theorem 2.1
• proof
• Definition 1: Parking Function Shuffle, p. 129 diaconis2017probabilizing