The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes

We give a new interpretation of the $\nu$-Tamari lattice of Préville-Ratelle and Viennot in terms of a rotation lattice of $\nu$-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, \mbox{tree-like} tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of $\nu$-trees, and show that the Hasse diagram of the $\nu$-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.