Count of Binary Operations on Set/Sequence

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

Let $N$ denote the number of possible different binary operations that can be applied to $S$:


 * $N = n^{\paren {n^2} }$

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

$\begin{array} {c|rr} n & n^2 & N = n^{\paren {n^2} } \\ \hline 1 & 1 & 1 \\ 2 & 4 & 16 \\ 3 & 9 & 19 \ 683 \\ 4 & 16 & 4 \ 294 \ 967 \ 296 \\ \end{array}$

There are still only $4$ elements in a set, and already there are over $4$ thousand million different possible algebraic structures.