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The Strip-Decomposition of m-Dyck Paths

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 $n\leq 3$, 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.