On the Number of Matroids on a Finite Set

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 − 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 subcollection of matroids. Two new tables are also presented, giving the number of paving matroids on at most eight elements. 1. Definitions Let Sn := {1, . . . , n}. A matroid M is a pair M(Sn,B) such that B is a non-empty collection of subsets of Sn satisfying the following basis-exchange property: ∀ X,Y ∈ B, ∀x ∈ X − Y,∃y ∈ Y −X s.t X − {x} ∪ {y} ∈ B. Sn is called the ground set of the matroid and B the collection of bases. A consequence of this axiom is that all sets in B have the same cardinality, called the rank of the matroid. The collection of independent sets of M is I(M) = {I : I ⊆ B,B ∈ B(M)}. An element x ∈ Sn is called a loop if {x} 6∈ I(M). If for every pair of distinct elements {x, y} ⊆ Sn there exists I ∈ I(M) s.t. I ⊇ {x, y}, then M is called a simple. LetMkr (Sn) be the class of rank-r matroids on Sn with all k-element sets independent, well defined for all 0 6 k 6 r 6 n. We say two matroids M1(Sn,B1) and M2(Sn,B2) are isomorphic if there exists a permutation π of Sn such that B2 = {π(B) : B ∈ B1}. Let M k r (Sn) be the corresponding class of non-isomorphic matroids (i.e. different matroids.) The collections M0r(Sn),M1r(Sn) andM2r(Sn) identify the classes of rank-rmatroids, loopless matroids and simple matroids, respectively, on Sn. Mr−1 r (Sn) is the class of rank-r paving matroids on Sn and the single matroid in Mrr(Sn) is the uniform rank-r matroid on n elements. Note that Mi+1 r (Sn) ⊂ Mir(Sn) for all 0 6 i < r. Given M ∈M0r(Sn) and X ⊆ Sn, the restriction of M to X, denoted M |X, is the matroid with independent sets {I ∩ X : I ∈ I(M)}. The dual of M is the matroid M⋆ ∈ M0n−r(Sn) with B(M⋆) := {Sn − B : B ∈ B(M)}. The rank of A ⊆ Sn is defined by r(A) := max{|X| : X ⊆ A,X ∈ I(M)}. We call A ⊆ Sn a flat if r(A∪{x}) = r(A)+1 for all x ∈ Sn−A. We denote by Fk(M) the collection of rank-k flats of M and the numbers Wk(M) := Supported by EC’s Research Training Network ‘Algebraic Combinatorics in Europe’, grant HPRN-CT-2001-00272 while the author was at Università di Roma Tor Vergata, Italy. 1