In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of ν-Tamari lattices. In our framework, the main role of "Catalan objects" is played by (I, J)-trees: bipartite trees associated to a pair (I, J) of finite index sets that stand in simple bijection with lattice paths weakly above a lattice path ν = ν(I, J). Such trees label the maximal simplices of a triangulation whose dual polyhedral complex gives a geometric realization of the ν-Tamari lattice introduced by Prévile-Ratelle and Viennot. In particular, we obtain geometric realizations of m-Tamari lattices as polyhedral subdivisions of associahedra induced by an arrangement of tropical hyperplanes, giving a positive answer to an open question of F. Bergeron.The simplicial complex underlying our triangulation endows the ν-Tamari lattice with a full simplicial complex structure. It is a natural generalization of the classical simplicial associahedron, alternative to the rational associahedron of Armstrong, Rhoades and Williams, whose h-vector entries are given by a suitable generalization of the Narayana numbers.Our methods are amenable to cyclic symmetry, which we use to present type B analogues of our constructions. Notably, we define a partial order that generalizes the type B Tamari lattice, introduced independently by Thomas and Reading, along with corresponding geometric realizations. 12 3.1. Flips and the (I, J)-Tamari lattice 132010 Mathematics Subject Classification. 05E45, 05E10, 52B22.

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 order, we provide a simple description of the lattice property. We also show that the ν-Tamari lattice 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.

This article empirically models a strategic game of technology adoption of shuttle train grain elevators with information on location of the firm and its competitors. A spatial econometric model illustrates the role of spatial interdependence of rivals' decisions as well as agronomic and competitive variables on discrete adoption decisions. The analysis assesses equilibria conditions that characterize technology adoption, in this case of shuttle train adoption, and the results provide an explanation of shuttle train adoption decisions in the grain handling industry in which spatial competition is critical. Copyright 2005, Oxford University Press.