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## Number of binary operations

Combinatorics
Grigorios Kostakos
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### Number of binary operations

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

NOTE: I don't have a solution.
Grigorios Kostakos
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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.