On the Lucky and Displacement Statistics of Stirling Permutations

On the Lucky and Displacement Statistics of Stirling Permutations

Laura Colmenarejo
Department of Mathematics
North Carolina State University
Raleigh, NC 27685
USA

Jennifer Elder
Department of Computer Science, Mathematics and Physics
Missouri Western State University
St. Joseph, MO 64507
USA

Kimberly J. Harry
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI 53211
USA

Dorian Smith
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA

Aleyah Dawkins
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
USA

Pamela E. Harris
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
Milwaukee, WI 53211
USA

Selvi Kara
Department of Mathematics
Bryn Mawr College
Bryn Mawr, PA 19010
USA

Bridget Eileen Tenner
Department of Mathematical Sciences
DePaul University
Chicago, IL 60614
USA

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 of the lucky statistic, both of which are enumerated by important integer sequences: 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 a selection of results for luck that lies between these two extremes. Further, we establish that the displacement of all the Stirling permutations 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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Received April 19 2024; revised versions received April 21 2024; August 9 2024. Published in Journal of Integer Sequences, August 9 2024.

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