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We give new interpretations of the ν-Tamari lattice of Préville-Ratelle and Viennot. First, we describe it as a rotation lattice of ν-trees, which uncovers the relation with known combinatorial objects such as tree-like tableaux and north-east fillings. Then, using a formulation in terms of bracket vectors of ν-trees and componentwise ∗Partially supported by the project “Austria/France Scientific & Technological Cooperation”, which is co-financed by the Austrian Federal Ministry of Science, Research and Economy BMWFW (Project No. FR 10/2018) and by the French Ministry of Foreign Affairs and International Development (PHC Amadeus 2018 Project No. 39444WJ); and by the Austrian Science Foundation FWF, grant F 5008-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. †Also supported by the “Austria/France Scientific & Technological Cooperation” project (Project No. FR 10/2018 and PHC Amadeus 2018 Project No. 39444WJ), as well as by the grant ANR-17-CE400018 of the French National Research Agency ANR (project CAPPS), and the program PEPS Jeunes Chercheur-e-s from the INSMI (CNRS). the electronic journal of combinatorics 27(1) (2020), #P1.14 1 order, we provide a simple description of the lattice property. We also show that the ν-Tamari is isomorphic to the increasing-flip poset of a suitably chosen subword complex, and settle a special case of Rubey’s lattice conjecture concerning the poset of pipe dreams defined by chute moves. Finally, this point of view generalizes to multi ν-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra. Mathematics Subject Classifications: 06A07, 05E45, 05E10, 05A05, 05A19

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