The Strip-Decomposition of m-Dyck Paths

Abstract

The $m$-Tamari lattices ${\mathcal{T}}_n^{(m)}$, introduced by Bergeron and Pr{é}ville-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 ${\mathcal{T}}_n$. While for ${\mathcal{T}}_n$ many combinatorial realizations are known, to present there is no further combinatorial realization of ${\mathcal{T}}_n^{(m)}$. 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 ${\mathcal{T}}_n^{(m)}$ as an induced subposet of the $m$-fold direct product of ${\mathcal{T}}_n$ with itself. We are able to prove this conjecture for $n\le 3$, and provide necessary conditions for $m$-tuples of Dyck paths to belong to this realization. However, for $n\ge 5$, no sufficient condition is known.

arXiv commentThe results of this paper are subsumed by arxiv:1312.2520, and it will therefore not be published