Many 2-Level Polytopes from Matroids

The family of 2-level matroids, that is, matroids whose base polytope is 2-level, has been recently studied and characterized by means of combinatorial properties. 2-Level matroids generalize series-parallel graphs, which have been already successfully analyzed from the enumerative perspective. We bring to light some structural properties of 2-level matroids and exploit them for enumerative purposes. Moreover, the counting results are used to show that the number of combinatorially non-equivalent (n - 1)-dimensional 2-level polytopes is bounded from below by c.n(-5/2). p(-n), where c approximate to 0.03791727 and p(-1) approximate to 4.88052854.