Connecting $k$-Naples parking functions and obstructed parking functions via involutions

TL;DR

This paper explores the relationship between $k$-Naples parking functions and classical parking functions, establishing bijections and bounds that deepen understanding of parking function generalizations.

Contribution

It introduces a bijection between contained $k$-Naples parking functions and classical parking functions, and extends this to relate $k$-Naples functions to obstructed parking functions.

Findings

Contained $k$-Naples parking functions have the same cardinality as classical parking functions.

A bijection is established between $k$-Naples and classical parking functions.

An injection into obstructed parking functions provides an upper bound for $k$-Naples functions.

Abstract

Parking functions were classically defined for $n$ cars attempting to park on a one-way street with $n$ parking spots, where cars only drive forward. Subsequently, parking functions have been generalized in various ways, including allowing cars the option of driving backward. The set $PF_{n,k}$ of $k$-Naples parking functions have cars who can drive backward a maximum of $k$ steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult. In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained $k$-Naples parking functions, has been shown, with a non-bijective proof, to have the same cardinality as that of the set $PF_n$ of classical parking functions, independent of $k$. In this paper, we study $k$-Naples parking functions in the more general context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various parking function involutions to establish a bijection between the contained $k$-Naples parking functions and the classical parking functions, from which it can be deduced that the two sets have the same number of ties. Then we extend this bijection to inject the set of $k$-Naples parking functions into a certain set of obstructed parking functions, providing an upper bound for the cardinality of the former set.