Geodesic
Some properties of the parking function poset
The electronic journal of combinatorics, Tome 29 (2022) no. 4
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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.
DOI : 10.37236/10714
Classification : 05A18, 05A15, 06A07, 52B05
Mots-clés : noncrossing 2-partitions, Whitney numbers, shellability 
@article{10_37236_10714,
     author = {B\'er\'enice Delcroix-Oger and Matthieu Josuat-Verg\`es and Lucas Randazzo},
     title = {Some properties of the parking function poset},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {4},
     doi = {10.37236/10714},
     zbl = {1506.05023},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10714/}
}
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Bérénice Delcroix-Oger; Matthieu Josuat-Vergès; Lucas Randazzo. Some properties of the parking function poset. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/10714

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