The harmonic polytope

Abstract: 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 $(n!)^2(1+\frac12+\cdots+\frac1n)$ vertices and $3^n-3$ 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.