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Many 2-Level Polytopes from Matroids

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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 \(\rho ^{-1} \approx 4.88052854\).

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Acknowledgments

F. G. was supported by the DFG within the research training group Methods for Discrete Structures (GRK1408). J. R. was partially supported by the Spanish MICINN Grant MTM2011-22851, the FP7-PEOPLE-2013-CIG project CountGraph (Ref. 630749), the DFG within the research training group Methods for Discrete Structures (GRK1408), and the Berlin Mathematical School. The authors thank Raman Sanyal for inspiring discussions and for accurate reading of this paper. Francisco Santos and Günter Ziegler are also thanked for helpful comments and suggestions.

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  1. Institut für Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2, 14195, Berlin, Germany

    Francesco Grande

  2. Institut für Mathematik und Informatik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Juanjo Rué

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  1. Francesco Grande
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  2. Juanjo Rué
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Correspondence to Francesco Grande.

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Grande, F., Rué, J. Many 2-Level Polytopes from Matroids. Discrete Comput Geom 54, 954–979 (2015). https://doi.org/10.1007/s00454-015-9735-5

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  • DOI: https://doi.org/10.1007/s00454-015-9735-5

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