Some properties of the parking function poset

Bérénice Delcroix-Oger (1, 2) , Matthieu Josuat-Vergès (1, 3, 4) , Lucas Randazzo (5, 6)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 IMAG - Institut Montpelliérain Alexander Grothendieck
3 UPCité - Université Paris Cité
4 CNRS - Centre National de la Recherche Scientifique
5 LIGM - Laboratoire d'Informatique Gaspard-Monge
6 UPEM - Université Paris-Est Marne-la-Vallée

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. The Electronic Journal of Combinatorics, 2022, 29 (4), Research Paper P4.42. ⟨10.37236/10714⟩. ⟨hal-03174934v2⟩