Some properties of the parking function poset
Discrete Mathematics
2023-01-06 v1 Combinatorics
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.
Cite
@article{arxiv.2103.14468,
title = {Some properties of the parking function poset},
author = {Bérénice Delcroix-Oger and Matthieu Josuat-Vergès and Lucas Randazzo},
journal= {arXiv preprint arXiv:2103.14468},
year = {2023}
}