The harmonic polytope

We study the harmonic polytope, which arose in Ardila, Denham, and Huh’s work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a (2n−2)-dimensional polytope with (n!)2(1+12+⋯+1n) vertices and 3n−3 facets. We also give a formula for its volume: it is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.