Undesired parking spaces and contractible pieces of the noncrossing partition link

TL;DR

This paper proves a conjecture that certain subcomplexes of the noncrossing partition link, related to parking functions, are contractible, using combinatorial topology and noncrossing hypertrees.

Contribution

It confirms the conjecture that collections of simplices determined by undesired parking spaces form contractible subcomplexes in the noncrossing partition link.

Findings

Confirmed contractibility of specific subcomplexes

Connected parking functions with topological properties

Applied hypertree theory to noncrossing partitions

Abstract

There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by combining the fact that the star of a simplex in a flag complex is contractible with the second author's theory of noncrossing hypertrees.