2023/12/28 by Ari Cruz, Pamela E. Harris, Cruz, Ari +13 · 2 citations
Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2312.16786
openalex publication_date 2023/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recall that α=(a1,a2,…,an)∈[n]n is a parking function if its nondecreasing rearrangement β=(b1,b2,…,bn) satisfies bi≤ i for all 1≤ i≤ n. In this article, we study parking functions based on their ascents (indices at which aiai+1), and ties (indices at which ai=ai+1). By utilizing multiset Eulerian polynomials, we give a generating function for the number of parking functions of length n with i descents. We present a recursive formula for the number of parking functions of length n with descents at a specified subset of [n-1]. We establish that the number of parking functions of length n with descents at I⊂[n-1] and descents at J=\n-i:i∈ I\ are equinumerous. As a special case, we show that the number of parking functions of length n with descents at the first k indices is given by f(n, n-k-1)=(1)/(n)\binomnk\binom2n-kn-k-1. We prove this by bijecting to the set of standard Young tableaux of shape ((n-k)2,1k), which are enumerated by f(n,n-k-1). We also study peaks of parking functions, which are indices at which ai-1ai+1. We show that the set of parking functions with no peaks and no ties is enumerated by the Catalan numbers. We conclude our study by characterizing when a parking function is uniquely determined by their statistic encoding; a word indicating what indices in the parking function are ascents, descents, and ties. We provide open problems throughout.