The Topology of the $m$-Tamari Lattices

Abstract

The $m$-Tamari lattices ${\mathcal{T}}_n^{(m)}$ were recently introduced by Bergeron and Préville-Ratelle as posets on $m$-Dyck paths, and it was shown by Bousquet-Mélou, Fusy and Préville-Ratelle that these lattices form intervals in the classical Tamari lattice ${\mathcal{T}}_{nm}$. It follows from a theorem by Björner and Wachs and a basic property of EL-shellable posets, that the $m$-Tamari lattices are EL-shellable. In this article, we define a new EL-labeling of the $m$-Tamari lattices completely in terms of $m$-Dyck paths. With the help of this labeling, we compute the values of the Möbius function of ${\mathcal{T}}_n^{(m)}$, and we characterize the intervals of ${\mathcal{T}}_n^{(m)}$ according to their topological properties.

11 pages, 3 figures. This article provides a new proof of the known fact that the m-Tamari lattices are EL-shellable, as well as to some topological consequences. Final version

References in corpus (1)

• The number of intervals in the m-Tamari lattices

Cited by in corpus (3)

• The representation of the symmetric group on m-Tamari intervals
• Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices
• Three Fuss-Catalan posets in interaction and their associative algebras