# How many pairs of permutations without fixed points

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 $$\sigma_1:=\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}$$ $$\sigma_2:=\begin{pmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \end{pmatrix}$$

• I do know the number of elements in Sn – UJung Jul 21 '18 at 13:47
• If $\mu$ and $\sigma$ are permutations without fixed points isn't the condition $\mu(\sigma(i))\neq\sigma(i)$ automatic? Otherwise $\sigma(i)$ would be a fixed point of $\mu$, no? – Jyrki Lahtonen Jul 21 '18 at 13:48
• For the number of permutations without fixed points see this thread. As you only seem to need both $\mu$ and $\sigma$ not to have any fixed points then that answers the question (but not in an entirely helpful form). – Jyrki Lahtonen Jul 21 '18 at 13:53
• See also here. – Jyrki Lahtonen Jul 21 '18 at 13:54
• Thanks, @joriki. No problem, really. I was expecting this edit since I saw Joffan and Ross Millikan comment. My interpretation didn't match with the single given data point. I only noticed that too late, i.e. after posting the answer. – Jyrki Lahtonen Jul 21 '18 at 19:19