Count of Binary Operations with Fixed Identity/Sequence
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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).