I'm interested in counting structures that satisfy certain constraints up to isomorphism. For example, I might want to know how many clutters there are on $n$ vertices.
The only way I can think to do this is by brute force---e.g. by picking some larger class of things which is easier to enumerate, and then throwing out examples that violate constraints (or are isomorphic to other examples). I'm sure there are better, more efficient techniques, but I know nothing about this area of mathematics.
I've been looking at using something like Alg for this---apparently it can count "finite models of single-sorted first-order theories." So my question is, what kinds of structures are these? I see there is a long list of examples provided with Alg and on wikipedia (list of first order theories). But if I have a structure (like the clutters I mentioned above), is it easy to tell whether there is a "single-sorted first-order theory" it is a model of?
If this is a very easy question given knowledge of some part of mathematics (model theory?) then I would interested in any reference recommendations you may have.