Count of Commutative Binary Operations with Fixed Identity/Sequence

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

Let $x \in S$.

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


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

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

$\begin {array} {c|cr} n & \dfrac {n \paren {n - 1} } 2 & n^{\frac {n \paren {n - 1} } 2} \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 3 & 27 \\ 4 & 6 & 4 \ 096 \\ 5 & 10 & 9 \ 765 \ 625 \\ \end {array}$