Geodesic
On the Number of Matroids of a Finite Set
Séminaire lotharingien de combinatoire, Tome 51 (2004-2005)

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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. 

@article{SLC_2004-2005_51_a6,
     author = {Mark Dukes},
     title = {On the {Number} of {Matroids} of a {Finite} {Set}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {51},
     year = {2004-2005},
     url = {http://geodesic.mathdoc.fr/item/SLC_2004-2005_51_a6/}
}
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Mark Dukes. On the Number of Matroids of a Finite Set. Séminaire lotharingien de combinatoire, Tome 51 (2004-2005). http://geodesic.mathdoc.fr/item/SLC_2004-2005_51_a6/