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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.03280 |
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| _version_ | 1866913256070709248 |
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| author | Colmenarejo, Laura Dawkins, Aleyah Elder, Jennifer Harris, Pamela E. Harry, Kimberly J. Kara, Selvi Smith, Dorian Tenner, Bridget Eileen |
| author_facet | Colmenarejo, Laura Dawkins, Aleyah Elder, Jennifer Harris, Pamela E. Harry, Kimberly J. Kara, Selvi Smith, Dorian Tenner, Bridget Eileen |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Lucky and Displacement Statistics of Stirling Permutations Colmenarejo, Laura Dawkins, Aleyah Elder, Jennifer Harris, Pamela E. Harry, Kimberly J. Kara, Selvi Smith, Dorian Tenner, Bridget Eileen Combinatorics 05A05, 05A15 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. |
| title | On the Lucky and Displacement Statistics of Stirling Permutations |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2403.03280 |