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Séminaire Lotharingien de Combinatoire, 84B.59 (2020), 12 pp.

# Bérénice Delcroix-Oger, Matthieu Josuat-Vergès and Lucas Randazza

# Some Properties of the Parking Function Poset

**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.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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