Geodesic
The Canonical Join Complex of the Tamari Lattice
Séminaire lotharingien de combinatoire, 80B (2018)

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In this paper, we study a simplicial complex on the elements of the Tamari lattice in types A and B called the canonical join complex. The canonical join representation of an element w in a lattice L is the unique lowest expression VA for w. We abuse notation and also say that the set A is a canonical join representation (when we mean VA is a canonical join representation). The collection of all such subsets is an abstract simplicial complex called the canonical join complex of L.

We realize the canonical join complex of the Tamari lattice as a complex of noncrossing arc diagrams, give a shelling order on its facets, and show that it is homotopy equivalent to a wedge of Catalan-many spheres. 

@article{SLC_2018_80B_a74,
     author = {Emily Barnard},
     title = {The {Canonical} {Join} {Complex} of the {Tamari} {Lattice}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a74/}
}
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Emily Barnard. The Canonical Join Complex of the Tamari Lattice. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a74/