Some Enumerations for Parking Functions

Abstract

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