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:


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

Proof
From Count of Binary Operations on Set, the total number of operations is $n^{\\paren {n^2} }$.

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

So the total number of operations without an identity is:


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

Hence the result.