Some properties of the parking function poset

Bérénice Delcroix-Oger (1) , Matthieu Josuat-Vergès (1, 2) , Lucas Randazzo (3)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 CNRS - Centre National de la Recherche Scientifique
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Bérénice Delcroix-Oger, Matthieu Josuat-Vergès, Lucas Randazzo. Some properties of the parking function poset. 2021. ⟨hal-03174934v1⟩