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

TL;DR

This paper introduces a new non-recursive formula for counting $k$-Naples parking functions using permutation subsequences and a novel $k$-Naples area statistic, extending classical parking function enumeration.

Contribution

It provides a new explicit enumeration formula for $k$-Naples parking functions based on permutation fibers, differing from previous recursive approaches.

Findings

Derived a non-recursive enumeration formula for $k$-Naples parking functions.

Connected the $q$-analog of the formula to a new $k$-Naples area statistic.

Established a recurrence relation for the generating function of the fiber sizes.

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 $0\leq k\leq n-1$, we define a function $\varphi_k$ 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 $\varphi_k$ 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 $\varphi_0$, 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 $\texttt{area}_k$ and call the $k$-Naples area statistic, the specialization of which to $k=0$ gives the $\texttt{area}$ statistic on parking functions.