The Topology of the m-Tamari Lattices

2012/01/10 by Mühle, Henri
#05E99 (Secondary) #06A07 (Primary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1201.2020

Abstract

The m-Tamari lattices 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 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 T_n^(m), and we characterize the intervals of T_n^(m) according to their topological properties.

Related

• The Strip-Decomposition of m-Dyck Paths
• Tamari lattices and parking functions: proof of a conjecture of F. Bergeron
• A partial order on Motzkin paths
• An extension of Tamari lattices
• Linear Intervals in the Tamari and the Dyck Lattices and in the alt-Tamari Posets