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On the number of matroids on a finite set

Dukes, W.M.B. (2004) On the number of matroids on a finite set. Séminaire Lotharingien de Combinatoire, 51. ISSN 1286-4889

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In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly improved, lower bound on the number of rank-r matroids on n elements when n=2m-1. We also prove an adjacent result showing the point-lines-planes conjecture to be true if and only if it is true for a special sub-collection of matroids. Two new tables are also presented, giving the number of paving matroids on at most eight elements.