Count of 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 binary operations which have an identity element that can be applied to $S$:

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


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

$\begin{array} {c|cr} n & \paren {n - 1}^2 + 1 & n^{\paren {n - 1}^2 + 1}\\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 5 & 243 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$

This sequence is A090602 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).