= Solution
Up to <matroid isomorphism>, the numbers for $n=1,2,3$ are \b[$2,4,8$]. A <matroid isomorphism> may relabel elements but preserves which <subsets> are <matroid independent sets>.
For $n=1$, the <loop of a matroid> and the <nonloop> are different types. For $n=2$, there is one type each at <rank of a matroid> zero and two, and two types at <rank of a matroid> one, distinguished by having one or two <nonloops>. Thus there are four.
For $n=3$, the seven labelled examples of <rank of a matroid> one reduce to three types, distinguished by having one, two or three <nonloops>. The seven labelled examples of <rank of a matroid> two also reduce to three types: one <loop of a matroid> and two independent elements; no <matroid loops> and one pair of <parallel elements of a matroid>; or the <uniform matroid> $U_{2,3}$. Different choices of labels within each description are isomorphic. Together with the unique types of <rank of a matroid> zero and three, this gives $1+3+3+1=8$. The <rank of a matroid>, number of <matroid loops> and presence of <parallel elements of a matroid> distinguish every type just listed.
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