Statistics on $\ell$-interval parking functions
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arXiv
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| Auteurs principaux: | , , , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915380577959936 |
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| author | Celano, Kyle Elder, Jennifer Hadaway, Kimberly P. Harris, Pamela E. Martin, Jeremy L. Priestley, Amanda Udell, Gabe |
| author_facet | Celano, Kyle Elder, Jennifer Hadaway, Kimberly P. Harris, Pamela E. Martin, Jeremy L. Priestley, Amanda Udell, Gabe |
| contents | The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among our results, we enumerate $\ell$-interval parking functions with respect to statistics such as inversion, displacement, and major index. We show that $1$-interval parking functions with fixed displacement exhibit a cyclic sieving phenomenon. We give closed formulas for the number of $1$-interval parking functions with a fixed number of inversions. We prove that a well-known bijection of Foata preserves the set of $\ell$-interval parking functions exactly when $\ell\leq 2$ or $\ell\geq n-2$, which implies that the inversion and major index statistics are equidistributed in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistics on $\ell$-interval parking functions Celano, Kyle Elder, Jennifer Hadaway, Kimberly P. Harris, Pamela E. Martin, Jeremy L. Priestley, Amanda Udell, Gabe Combinatorics Primary 05A05, Secondary 05A15, 05A19 The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among our results, we enumerate $\ell$-interval parking functions with respect to statistics such as inversion, displacement, and major index. We show that $1$-interval parking functions with fixed displacement exhibit a cyclic sieving phenomenon. We give closed formulas for the number of $1$-interval parking functions with a fixed number of inversions. We prove that a well-known bijection of Foata preserves the set of $\ell$-interval parking functions exactly when $\ell\leq 2$ or $\ell\geq n-2$, which implies that the inversion and major index statistics are equidistributed in these cases. |
| title | Statistics on $\ell$-interval parking functions |
| topic | Combinatorics Primary 05A05, Secondary 05A15, 05A19 |
| url | https://arxiv.org/abs/2507.07243 |