Enumerating $k$-Naples Parking Functions Through Catalan Objects

TL;DR

This paper generalizes parking functions to $k$-Naples parking functions with backward movement, establishing bijections with Catalan objects to enumerate both ascending and descending variants.

Contribution

It introduces a new class of parking functions allowing backward moves and extends classical bijections to enumerate these functions.

Findings

Established bijections with Dyck paths, binary trees, and non-crossing partitions.

Derived enumeration formulas for $k$-Naples parking functions.

Showed differences in counts between ascending and descending variants.

Abstract

This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitions. Using these bijections, we enumerate both ascending and descending $k$-Naples parking functions.