I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the definition for exponentiation is:
$b^n = \underbrace{b \times \cdots \times b}_n$
And the definition for factorial is:
$n ! = n \times...\times 1 $
So, for both exponentiation ($x^n$) and factorial ($n!$) shouldn't $n$ be the arity? Since we need to perform $n$ multiplications in order to evaluate it? With one exception for $0!=1$ which may be a unary operation.