Hochschild polytopes

The (m, n)-multiplihedron is a polytope whose faces correspond to m-painted n-trees. Deleting certain inequalities from its facet description, we obtain the (m, n)Hochschild polytope whose faces correspond to m-lighted n-shades. Moreover, there is a natural shadow map from m-painted n-trees to m-lighted n-shades, which defines a meet semilattice morphism of rotation lattices. In particular, when m = 1, our Hochschild polytope is a deformed permutahedron realizing the Hochschild lattice.