# Count of Commutative Binary Operations on Set/Sequence

## Theorem

Let $S$ be a set whose cardinality is $n$.

Let $N$ denote the number of different commutative binary operations that can be applied to $S$:

$N = n^{\frac {n \paren {n + 1} } 2}$

The sequence of $N$ for each $n$ begins:

$\begin{array} {c|cr} n & \dfrac {n \paren {n + 1} }2 & n^{\frac {n \paren {n + 1} } 2} \\ \hline 1 & 1 & 1 \\ 2 & 3 & 8 \\ 3 & 6 & 729 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$

and so on.