Combinatorial Theory

The harmonic polytope and the bipermutahedron are two related polytopes that arose in the Lagrangian geometry of matroids. We study the bipermutahedron. We show that it is a simple polytope whose faces are in bijection with the vertex-labeled and edge-labeled multigraphs with no isolated vertices; the generating function for its \(f\)-vector is a simple evaluation of the three variable Rogers-Ramanujan function. We introduce the biEulerian polynomial, which counts bipermutations according to their number of descents, and equals the \(h\)-polynomial of the bipermutahedral fan. We construct a unimodular triangulation of the product \(\Delta \times \cdots \times \Delta\) of triangles that is combinatorially equivalent to (the triple cone over) the bipermutahedral fan. Ehrhart theory then gives us a formula for the biEulerian polynomial, which we use to show that this polynomial is real-rooted and that the \(h\)-vector of the bipermutahedral fan is log-concave and unimodal. We describe all the deformations of the bipermutahedron; that is, the ample cone of the bipermutahedral toric variety. We prove that among all polytopes in this family, the bipermutahedron has the largest possible symmetry group. Finally, we show that the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.

Mathematics Subject Classifications: 52B20, 52B05, 05A15

Keywords: Polytope, bipermutahedron, bipermutations, descents, \(f\)-vector, \(h\)-vector, unimodular triangulation, Ehrhart polynomial, real-rooted polynomial, deformation cone