Count of Binary Operations Without Identity

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

The number $$N$$ of possible different binary operations which do not have an identity element that can be applied to $$S$$ is given by:


 * $$N = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$$

Proof
From Count of Binary Operations on a Set, the total number of operations is $$n^{\left({n^2}\right)}$$.

From Count of Binary Operations with Identity, the total number of operations with an identity is $$n^{\left({n-1}\right)^2 + 1}$$.

So the total number of operations without an identity is:

$$n^{\left({n^2}\right)} - n^{\left({n-1}\right)^2 + 1} = n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)$$

Hence the result.

Comment
The number grows rapidly with $$n$$:

$$\begin{array} {c||r|r|r} n & n^{\left({n^2}\right)} & n^{\left({n-1}\right)^2 + 1} & n^{\left({\left({n-1}\right)^2 + 1}\right)} \left({n^{2 \left({n-1}\right)} - 1}\right)\\ \hline 1 & 1 & 1 & 0 \\ 2 & 16 & 4 & 12 \\ 3 & 19 \ 683 & 243 & 19 \ 440 \\ 4 & 4 \ 294 \ 967 \ 296 & 1 \ 048 \ 576 & 4 \ 293 \ 918 \ 720\\ \end{array}$$

This sequence has not yet been defined in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008)