Count of Commutative Binary Operations with Identity/Sequence

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Theorem

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

Let $N$ denote the number of different commutative binary operations which have an identity element that can be applied to $S$:

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


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

$\begin{array} {c|cr} n & \dfrac {n \paren {n - 1} } 2 + 1 & n^{\frac {n \paren {n - 1} } 2 + 1} \\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 4 & 81 \\ 4 & 7 & 16 \ 384 \\ 5 & 11 & 48 \ 828 \ 125 \\ \end{array}$

This sequence is A090599 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).