How many distinct commutative binary operations can be defined in a set of $n$ elements?
NOTE: I don't have a solution.
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Number of binary operations
 Grigorios Kostakos
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Number of binary operations
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Re: Number of binary operations
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.
Of course there are \(n^{\binom{n}{2} + n} = n^{\binom{n+1}{2}.}\) such choices.