On the Numbers of Bases and Circuits in Simple Binary Matroids

provided again that M is simple and binary. The reason for restricting attention here to simple binary matroids is that expression (1.1) need not hold for arbitrary simple matroids or even for loopless graphic matroids. To see, this, consider, for example, the uniform matroids U 2 ,m form~ 6 and U 1,n for n ~ 4. We observe here that inequality (1.3) is a sharper bound than expression (1.2) unless M is the direct sum of a free matroid and a matroid of rank less than six. We have included expression (1.2) because it is used in the proof of inequality (1.3). Indeed, without it, one obtains the weaker bound