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Séminaire Lotharingien de Combinatoire, B51g (2004), 12 pp.

# Mark Dukes

# On the Number of Matroids of a Finite Set

**Abstract.**
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*=2^{m}-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.

Received: February 4, 2004.
Revised: December 1, 2004; December 9, 2004.
Accepted: December 9, 2004.

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