Enumerating matroids of fixed rank
Rudi Pendavingh, Jorn van der Pol

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Abstract
It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that , where denotes the number of matroids on a fixed groundset of size , and the number of sparse paving matroids. In an earlier paper, we showed that . The bounds that we used for that result were dominated by matroids of rank . In this paper we consider the relation between the number of sparse paving matroids and the number of matroids on a fixed groundset of size of fixed rank . In particular, we show that whenever , by giving asymptotically matching upper and lower bounds. Our upper bound on relies heavily on the theory of matroid erections as developed by Crapo and Knuth, which we use to encode any matroid as a stack of paving matroids. Our best…
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