The Strip-Decomposition of m-Dyck Paths

2013/08/22 by Henri Mühle, Mühle, Henri
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.1308.4804

Abstract

The m-Tamari lattices T_n^(m), introduced by Bergeron and Préville-Ratelle, are defined as a poset of m-Dyck paths equipped with the generalized rotation order, and constitute a Fuss-Catalan generalization of the classical Tamari lattices T_n. While for T_n many combinatorial realizations are known, to present there is no further combinatorial realization of T_n^(m). In this article, we introduce a certain decomposition of m-Dyck paths into m-tuples of Dyck paths, and after a certain modification of these m-tuples, we conjecture that the resulting m-tuples of Dyck paths realize T_n^(m) as an induced subposet of the m-fold direct product of T_n with itself. We are able to prove this conjecture for n≤ 3, and provide necessary conditions for m-tuples of Dyck paths to belong to this realization. However, for n≥ 5, no sufficient condition is known.

Related

• The Topology of the m-Tamari Lattices
• An extension of Tamari lattices
• A partial order on Motzkin paths
• A remark on s-torsion pairs and on the lattice of Dyck paths
• Counting generalized Dyck paths