Counting $k$-Naples parking functions through permutations and the $k$-Naples area statistic

Abstract

We recall that the $k$-Naples parking functions of length $n$ (a generalization of parking functions) are defined by requiring that a car which finds its preferred spot occupied must first back up a spot at a time (up to $k$ spots) before proceeding forward down the street. Note that the parking functions are the specialization of $k$ to $0$. For a fixed $0\le k\le n-1$, we define a function ${\phi }_k$ which maps a $k$-Naples parking function to the permutation denoting the order in which its cars park. By enumerating the sizes of the fibers of the map ${\phi }_k$ we give a new formula for the number of $k$-Naples parking functions as a sum over the permutations of length $n$. We remark that our formula for enumerating $k$-Naples parking functions is not recursive, in contrast to the previously known formula of Christensen et al [CHJ+20]. It can be expressed as the product of the lengths of particular subsequences of permutations, and its specialization to $k=0$ gives a new way to describe the number of parking functions of length $n$. We give a formula for the sizes of the fibers of the map ${\phi }_0$, and we provide a recurrence relation for its corresponding logarithmic generating function. Furthermore, we relate the $q$-analog of our formula to a new statistic that we denote ${\text{area}}_k$ and call the $k$-Naples area statistic, the specialization of which to $k=0$ gives the $\text{area}$ statistic on parking functions.

17 pages, 2 figures, 1 table

References in corpus (1)

• On Increasing and Invariant Parking Sequences