Enumerating matroids of fixed rank

Rudi Pendavingh, Jorn van der Pol

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It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that s(n)∼m(n) , where m(n) denotes the number of matroids on a fixed groundset of size n , and s(n) the number of sparse paving matroids. In an earlier paper, we showed that logs(n)∼logm(n) . The bounds that we used for that result were dominated by matroids of rank r≈n/2 . In this paper we consider the relation between the number of sparse paving matroids s(n,r) and the number of matroids m(n,r) on a fixed groundset of size n of fixed rank r . In particular, we show that logs(n,r)∼logm(n,r) whenever r≥3 , by giving asymptotically matching upper and lower bounds.
Our upper bound on m(n,r) 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 result is obtained by relating to this stack of paving matroids an antichain that completely determines the matroid.
We also obtain that the collection of essential flats and their ranks gives a concise description of matroids.
Original languageEnglish
Number of pages18
Issue number1512.06655
Publication statusPublished - 21 Dec 2015


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