Some Enumerations for Parking Functions

TL;DR

This paper derives formulas, recurrence relations, and generating functions for various classes of parking functions, analyzing their asymptotic behavior and providing new enumerative insights into these combinatorial objects.

Contribution

It introduces new formulas and recurrence relations for counting specific parking functions and studies their asymptotic properties, expanding understanding of parking function enumeration.

Findings

Derived formulas for counting parking functions with constraints

Established recurrence relations for related sequences

Analyzed asymptotic behavior of these sequences

Abstract

In this paper, let $\mathcal{P}_{n,n+k;\leq n+k}$ (resp. $\mathcal{P}_{n;\leq s}$) denote the set of parking functions $\alpha=(a_1,...,a_n)$ of length $n$ with $n+k$ (respe. $n$)parking spaces satisfying $1\leq a_i\leq n+k$ (resp. $1\leq a_i\leq s$) for all $i$. Let $p_{n,n+k;\leq n+k}=|\mathcal{P}_{n,n+k;\leq n+k}|$ and $p_{n;\leq s}=|\mathcal{P}_{n;\leq s}|$. Let $\mathcal{P}_{n;\leq s}^l$ denote the set of parking functions $\alpha=(a_1,...,a_n)\in\mathcal{P}_{n;\leq s}$ such that $a_1=l$ and $p_{n;\leq s}^l=|\mathcal{P}_{n;\leq s}^l|$. We derive some formulas and recurrence relations for the sequences $p_{n,n+k;\leq n+k}$, $p_{n;\leq s}$ and $p_{n;\leq s}^l$ and give the generating functions for these sequences. We also study the asymptotic behavior for these sequences.