Let's say $S_n$ is the set of all permutations of {1,2,...,n} without fixed points.
Now what I want to find out is the number of pairs of permutations $(\sigma,\mu)$ where
$$\sigma, \mu \in S_n$$ and $$\mu(\sigma(i))\neq i$$ for every $i\in \text{{1,2,...,n}}$
For example, for the case n=3, only two pairs ($\sigma_1$,$\sigma_1$) and ($\sigma_2$,$\sigma_2$) satisfies all the conditions where $$\sigma_1:=\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}$$ $$\sigma_2:=\begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$