A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young’s Lattice

We present a unifying framework in which both the ν -Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the h^∗ -vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of Mészáros when ν is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.