For one element, the element is either a loop of a matroid or a nonloop, giving two choices. For two elements, rank of a matroid zero and two each give one possibility. At rank of a matroid one, the nonloops form any nonempty subset, giving more possibilities: total .
For three elements, classify by the total rank of a matroid. At rank of a matroid zero every element is a matroid loop; at rank of a matroid three every subset is a matroid independent set. At rank of a matroid one, choose any nonempty subset of nonloops; exactly the empty set and their singletons are matroid independent sets. This gives seven possibilities.
At rank of a matroid two, either exactly two elements are nonloops, giving three choices for the remaining loop of a matroid, or all three are nonloops. In the latter case, pairs that are matroid dependent sets define parallel elements of a matroid. There can be no such pair, giving the uniform matroid , or exactly one such pair, giving three choices. There cannot be exactly two such pairs: if and were matroid dependent sets but a matroid independent set, augmentation from to would fail. If all three pairs were matroid dependent sets, the rank of a matroid would be one. Hence rank of a matroid two gives examples.
Every listed possibility satisfies augmentation: the only nontrivial comparison on three elements is from a singleton to a two-element matroid independent set, and with at most one two-element matroid dependent set at least one addition works. The total is therefore .
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