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arXiv · 2507.07243

Statistics on $\ell$-interval parking functions

Kyle Celano·Jennifer Elder·Kimberly P. Hadaway·Pamela E. Harris·Jeremy L. Martin·Amanda Priestley·Gabe Udell
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Abstract

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.

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Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Jeremy L. Martin, Amanda Priestley, Gabe Udell. 2025-07-09. Statistics on $\ell$-interval parking functions. https://arxiv.org/abs/2507.07243

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