Some properties of the parking function poset

TL;DR

This paper explores the properties of a poset related to parking functions, providing enumerative, topological, and algebraic insights, including shellability, homology, and connections to well-known polytopes.

Contribution

It introduces new enumerative and topological results for the parking function poset, including shellability and homology computations, and links to associahedron and permutohedron.

Findings

Derived a formula for counting chains in the poset

Proved the poset is shellable

Computed the homology as a symmetric group representation

Abstract

In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with two well-known polytopes : the associahedron and the permutohedron.