A unifying framework for the $ν$-Tamari lattice and principal order ideals in Young's lattice

Abstract

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, Padrol and Sarmiento. 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.