On the Lucky and Displacement Statistics of Stirling Permutations

2024/03/05 by Laura Colmenarejo, Colmenarejo, Laura, Aleyah Dawkins +13 · 4 citations
Mathematics · #05A05 #05A15 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2403.03280

openalex publication_date 2024/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Stirling permutations are parking functions, and we investigate two parking function statistics in the context of these objects: lucky cars and displacement. Among our results, we consider two extreme cases: extremely lucky Stirling permutations (those with maximally many lucky cars) and extremely unlucky Stirling permutations (those with exactly one lucky car). We show that the number of extremely lucky Stirling permutations of order n is the Catalan number C_n, and the number of extremely unlucky Stirling permutations is (n-1)!. We also give some results for luck that lies between these two extremes. Further, we establish that the displacement of any Stirling permutation of order n is n², and we prove several results about displacement composition vectors. We conclude with directions for further study.

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