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A matroid isomorphism between $(E,\mathcal{I})$ and $(E^{\prime },{\mathcal{I}}^{\prime })$ is a bijection $\varphi$ such that $I\in \mathcal{I}$ if and only if $\varphi (I)\in {\mathcal{I}}^{\prime }$. It relabels the ground set without changing independence. The number of labelled matroids therefore differs from the number of types up to matroid isomorphism.

Up to matroid isomorphism, the numbers for $n=1,2,3$ are $2,4,8$. 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.

On labelled ground sets of sizes $1,2,3$, there are respectively $2,5,16$ matroids, or $2,4,8$ up to matroid isomorphism. At rank of a matroid one, a nonempty subset of nonloops specifies all matroid independent sets of cardinality one. On three labels at rank of a matroid two, the three types are one matroid loop with an independent pair, no matroid loops with one pair of parallel elements of a matroid, and all pairs independent with the triple dependent.