A Decomposition of Parking Functions by Undesired Spaces

TL;DR

This paper introduces a novel decomposition of parking functions based on omitted numbers, revealing structural properties and connections to various combinatorial objects.

Contribution

It presents a new way to decompose parking functions and analyzes the resulting posets, uncovering properties similar to the noncrossing partition lattice.

Findings

Posets share properties with the noncrossing partition lattice.

Intervals in the posets have a nice characterization.

The M"obius function of the posets is readily computable.

Abstract

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable M\"obius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.