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, 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.

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