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Combinatorica

, Volume 38, Issue 4, pp 955–985 | Cite as

On the Number of Bases of Almost All Matroids

  • Rudi Pendavingh
  • Jorn van der Pol
Article

Abstract

For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that
$$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$
for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a Uk,2k-minor, whenever kO(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity \(\geq\Omega\;{(\sqrt {log(n)})}\), and (4) do not arise as the truncation of another matroid.

Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.

Mathematics Subject Classification (2000)

05B35 05A16 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Eindhoven University of TechnologyEindhoventhe Netherlands
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada

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