Discrete & Computational Geometry

, Volume 54, Issue 4, pp 954–979 | Cite as

Many 2-Level Polytopes from Matroids

Article

Abstract

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 \(c \cdot n^{-5/2} \cdot \rho ^{-n}\), where \(c\approx 0.03791727 \) and \(\rho ^{-1} \approx 4.88052854\).

Keywords

Matroid theory 2-level polytopes Analytic combinatorics Asymptotic enumeration 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut für Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik und InformatikFreie Universität BerlinBerlinGermany

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