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

UJung
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  • I do know the number of elements in Sn – UJung Jul 21 '18 at 13:47
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    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
  • Do you perhaps mean $\mu(\sigma(i))\ne i$? That is, $\mu\sigma\in S_n$? – Joffan Jul 21 '18 at 14:03
  • Maybe what you want is $\mu(\sigma(i))\neq i$. Then it can't just be the derangements. – Ross Millikan Jul 21 '18 at 14:04
  • @Joffan, that actually makes sense. Otherwise the OP's example of $n=3$ doesn't fit! – Jyrki Lahtonen Jul 21 '18 at 14:13
  • UJung, please clarify. If you really only want $\mu(\sigma(i))\neq \sigma(i)$ then $(\sigma_1,\sigma_2)$ and $(\sigma_2,\sigma_1)$ also work! – Jyrki Lahtonen Jul 21 '18 at 14:14
  • I just changed the last condition! – UJung Jul 21 '18 at 18:38
  • If you change the question such that the change invalidates one of the existing correct answers, you should notify the author of the answer (by commenting under the answer). Otherwise you make it look as if they posted a wrong answer. (Alternatively, you can mark the edit as an edit in the question so that it's apparent that the question changed fundamentally after the answer was posted.) In fact, the best course of action if you notice that you accidentally asked the wrong question and it's already been answered would be to post a new question. – joriki Jul 21 '18 at 19:06
  • You write that $S_n$ is "a" set of permutations of ${1,2,\ldots,n}$ without fixed points. That's confusing in two respects. First, $S_n$ is the conventional notation for the set of all permutations of ${1,2,\ldots,n}$. Second, due to the indefinite article it's not clear precisely which set you mean. Do you mean that $S_n$ is the set of all permutations of ${1,2,\ldots,n}$ without fixed points? – joriki Jul 21 '18 at 19:11
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    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
  • @joriki, Oh thank you for the tip! – UJung Jul 21 '18 at 20:16
  • @JyrkiLahtonen, sorry for the confusion. – UJung Jul 21 '18 at 20:28
  • @joriki, yes, "the set of all permutations" will be correct. I've just changed. Thank you. – UJung Jul 21 '18 at 20:47

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