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Number of binary operations

Combinatorics
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Grigorios Kostakos
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Number of binary operations

#1

Post by Grigorios Kostakos » Mon Jan 18, 2016 4:15 am

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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Re: Number of binary operations

#2

Post by Demetres » Mon Jan 18, 2016 4:16 am

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.
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