A new approach to Naples parking functions through complete parking preferences

TL;DR

This paper introduces complete parking preferences to analyze Naples parking functions, providing new combinatorial characterizations and linking permutation-invariant cases to existing results with a novel approach.

Contribution

It defines complete parking preferences and characterizes Naples parking functions through these, offering a new perspective and alternative proof for permutation-invariant cases.

Findings

Characterization of Naples parking functions via complete subsequences

Equivalent description of permutation-invariant Naples parking functions

New combinatorial framework for analyzing parking functions

Abstract

Naples parking functions were introduced as a generalization of classical parking functions, in which cars are allowed to park backwards, by checking up to a fixed number of previous spots, before proceeding forward as usual. In this work we introduce the notion of a complete parking preference, through which we are able to give some information on the combinatorics of Naples parking functions. Roughly speaking, a complete parking preference is a parking preference such that, for any index $j$, there are more cars with preference at least $j$ than spots available from $j$ onward. We provide a characterization of Naples parking functions in terms of certain complete subsequences of them. As a consequence of this result we derive a characterization of permutation-invariant Naples parking functions which turns out to be equivalent to the one given by (Carvalho et al., 2021), but using a totally different approach (and language).