Statistics on $\ell$-interval parking functions

TL;DR

This paper studies $ ext{l}$-interval parking functions, providing enumeration formulas, exploring their combinatorial properties, and revealing symmetry phenomena such as cyclic sieving and equidistribution of statistics.

Contribution

It introduces enumeration formulas for $ ext{l}$-interval parking functions, analyzes their statistics, and establishes when certain bijections preserve these functions.

Findings

Enumeration formulas for $ ext{l}$-interval parking functions.

Cyclic sieving phenomenon for 1-interval parking functions with fixed displacement.

Conditions under which inversion and major index are equidistributed.

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.