Some Enumerations for Parking Functions

2008/06/03 by Po-Yi Huang, Jun Ma, Huang, Po-Yi +3
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #math.CO

paper · pdf · doi:10.48550/arxiv.0806.0424

arxiv created 2008/06/03 · openalex publication_date 2008/06/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, let P_{n,n+k;≤ n+k} (resp. P_{n;≤ s}) denote the set of parking functions α=(a₁,...,a_n) of length n with n+k (respe. n)parking spaces satisfying 1≤ a_i≤ n+k (resp. 1≤ a_i≤ s) for all i. Let p_{n,n+k;≤ n+k}=|P_{n,n+k;≤ n+k}| and p_{n;≤ s}=|P_{n;≤ s}|. Let P_{n;≤ s}^l denote the set of parking functions α=(a₁,...,a_n)\inP_{n;≤ s} such that a₁=l and p_{n;≤ s}^l=|P_{n;≤ s}^l|. We derive some formulas and recurrence relations for the sequences p_{n,n+k;≤ n+k}, p_{n;≤ s} and p_{n;≤ s}^l and give the generating functions for these sequences. We also study the asymptotic behavior for these sequences.

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