On the Lucky and Displacement Statistics of Stirling Permutations

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 \(n^2\), and we prove several results about displacement composition vectors. We conclude with directions for further study.