Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible Cayley tables, that is $n^{n^2}$ and trying to divide this figure by something. Now, what is this something? I guess it's related to $n!$ but my imagination stops here.
EDIT: I think, but might well be wrong, that the two following problems are equivalent restatements of the original problem.
graph-theoretical flavor: enumerate up to isomorphism all the bipartite graphs on the sets $A$ and $B$ such that $|A|=n^2$ and $|B|=n$ and every vertex in $A$ has degree one.
"ball-boxing" flavor: enumerate in how many ways one can distribute $n^2$ identical balls in $n$ identical boxes.
A (very complicated) formula computing the number of isomorphism classes of magmas of a given, finite order appears in this article by M.A. Harrison. It is derived by understanding the action of the symmetric group $S_n$ on $n\times n$ Cayley tables and using evaluations of cycle index polynomials to derive the formula. See also this OEIS entry for other references and further values.