The Strip-Decomposition of m-Dyck Paths

TL;DR

This paper introduces a new decomposition method for m-Dyck paths aiming to realize m-Tamari lattices as subposets of product lattices, providing partial proofs and conditions for small cases.

Contribution

It proposes a novel decomposition of m-Dyck paths and conjectures a combinatorial realization of m-Tamari lattices within product lattices, with proofs for small cases.

Findings

Proved the conjecture for n ≤ 3.

Provided necessary conditions for m-tuple realizations.

Identified open problems for n ≥ 5.

Abstract

The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$, introduced by Bergeron and Pr{\'e}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 $n\leq 3$, and provide necessary conditions for $m$-tuples of Dyck paths to belong to this realization. However, for $n\geq 5$, no sufficient condition is known.