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arXiv · 1201.2020

The Topology of the $m$-Tamari Lattices

Henri Mühle
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Abstract

The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$ 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 $\mathcal{T}_{nm}$. 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 $\mathcal{T}_{n}^{(m)}$, and we characterize the intervals of $\mathcal{T}_{n}^{(m)}$ according to their topological properties.

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Henri Mühle. 2015-06-10. The Topology of the $m$-Tamari Lattices. https://arxiv.org/abs/1201.2020

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