Tropical Catalan subdivisions

We revisit the associahedral subdivision of the Pitman-Stanley polytope to provide geometric realizations of the ν-Tamari lattice of Préville-Ratelle and Viennot (which generalizes the m-Tamari lattice) as the dual of a triangulation of a polytope, as the dual of a mixed subdivision and as the edge-graph of a polyhedral complex induced by a tropical hyperplane arrangement. The method generalizes to type Bn.