Enumerating Matroids and Linear Spaces

Abstract

We show that the number of linear spaces on a set of n points and the number of rank-3 matroids on a ground set of size n are both of the form (cn+o(n)) n 2 /6 , where c=e 3/2-3 (1+3)/2. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: there are exact formulas for enumeration of rank-1 and rank-2 matroids, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant r≥4 there are (e 1-r n+o(n)) n r-1 /r! rank-r matroids on a ground set of size n.