Count of Binary Operations with Fixed Identity

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

Let $x \in S$.

The number $N$ of possible different binary operations such that $x$ is an identity element that can be applied to $S$ is given by:


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

Proof
Let $S$ be a set such that $\card S = n$.

Let $x \in S$ be an identity element.

From Count of Binary Operations on Set, there are $n^{\paren {n^2} }$ binary operations in total.

We also know that $a \in S \implies a \circ x = a = x \circ a$, so all operations on $x$ are already specified.

It remains to count all possible combinations of the remaining $n - 1$ elements.

This is effectively counting the mappings $\paren {S \setminus \set x} \times \paren {S \setminus \set x} \to S$.

From Count of Binary Operations on Set, this is $n^{\paren {\paren {n - 1}^2} }$ structures with $x$ as the identity element.