The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes - Ceballos, Cesar; Padrol, Arnau; Sarmiento, Camilo (The Electronic Journal of Combinatorics, 2020)
We give a new interpretation of the u-Tamari lattice of Préville-Ratelle and Viennot in terms of a rotation lattice of u-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, tree-like tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of u-trees, and show that the Hasse diagram of the u-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi u-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.