Counting Finite Magmas

Given a non-negative integer $n$, we establish a formula for the number of finite magmas on a set with cardinality $n$ up to isomorphism. We then generalize the method to operations with arbitrary finite arity, which yields a corrected version of Harrison's formula. Moreover, we present the cycle index as a helpful tool for practical computations and, based on that, we give a suitable code in Sage with a few generated examples.