Number of binary operations

[Grigorios Kostakos]

How many distinct commutative binary operations can be defined in a set of $n$ elements?

NOTE: I don't have a solution.

[Demetres]

Let us denote by \(S =\{x_1,\ldots,x_n\}\) our set. We need to define all \(x_i \ast x_j\) for each \(1 \leqslant i,j \leqslant n\) and the only restrictions are that \(x_i \ast x_j = x_j \ast x_i\). So \(\ast\) is uniquely determined by the definitions of \( x_i \ast x_j\) for each pair \( (i,j) \) with \(1 \leqslant i \leqslant j \leqslant n\).

Of course there are \(n^{\binom{n}{2} + n} = n^{\binom{n+1}{2}.}\) such choices.