Count of unique algebras with one unary operation (on finite set)

Question

It's possible, that count of unique algebras (up to isomorphism) with one unary operation on set with $n$ elements is $2^n-1$?

For $n=1,2,3,4$ is this hypothesis true (I still have not verified it on the computer). Is there any counterexample, or idea for proof?

Thanks.

Answer 1

In brief: no. There are 19 algebras on 4 elements, and the pattern breaks down after that.

Instead of algebras with one unary operation, we might as well speak of nonisomorphic functions on a set of $n$ elements. The answer is then given by Theorem 6 of "The Number of Structures of Finite Relations" by Davis, R.L. A pdf is available here.