On framed triangulations of flow polytopes, the v-Tamari lattice and Young’s lattice

We study two combinatorially striking triangulations of a family of flow polytopes indexed by lattice paths v which we call the v-caracol flow polytopes. The first triangulation gives a geometric realization of the v-Tamari complex introduced by Ceballos, Padrol and Sarmiento, whose dual graph is the Hasse diagram of the v-Tamari lattice introduced by Préville-Ratelle and Viennot. The dual graph of the second triangulation is the Hasse diagram of the principal order ideal determined by v in Young’s lattice. We use the latter triangulation to show that the h*-vector of the v-caracol flow polytope is given by the v-Narayana numbers, extending the result of Mészáros when v is a staircase lattice path.