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 $$(n-1)$$(n-1)-dimensional 2-level polytopes is bounded from below by $$c \cdot n^{-5/2} \cdot \rho ^{-n}$$c·n-5/2·ρ-n, where $$c\approx 0.03791727 $$c≈0.03791727 and $$\rho ^{-1} \approx 4.88052854$$ρ-1≈4.88052854.

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