Count of 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 binary operations such that $x$ is an identity element that can be applied to $S$:

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

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

$\begin {array} {c|cr} n & \paren {n - 1}^2 & n^{\paren {\paren {n - 1}^2} } \\ \hline 1 & 0 & 1 \\ 2 & 1 & 2 \\ 3 & 4 & 81 \\ 4 & 9 & 262 \, 144 \\ \end{array}$

This sequence is A090603 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).