How many pairs of permutations without fixed points

Clash Royale CLAN TAG#URR8PPP

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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 $iin text1,2,...,n$

For example, for the case n=3, only two pairs ($sigma_1$,$sigma_1$) and ($sigma_2$,$sigma_2$) satisfies all the conditions where
$$sigma_1:=beginpmatrix 1 & 2 & 3 \ 2 & 3 & 1 endpmatrix$$
$$sigma_2:=beginpmatrix 1 & 2 & 3 \ 3 & 1 & 2 endpmatrix$$

abstract-algebra permutations

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edited Jul 21 at 20:47

asked Jul 21 at 13:44

UJung

112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I do know the number of elements in Sn
  – UJung
  Jul 21 at 13:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
  – Jyrki Lahtonen
  Jul 21 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 at 13:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See also here.
  – Jyrki Lahtonen
  Jul 21 at 13:54
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  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 at 19:19
  

  
  
  
  

 | 
show 10 more comments

up vote
2
down vote

favorite

1

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 $iin text1,2,...,n$

For example, for the case n=3, only two pairs ($sigma_1$,$sigma_1$) and ($sigma_2$,$sigma_2$) satisfies all the conditions where
$$sigma_1:=beginpmatrix 1 & 2 & 3 \ 2 & 3 & 1 endpmatrix$$
$$sigma_2:=beginpmatrix 1 & 2 & 3 \ 3 & 1 & 2 endpmatrix$$

abstract-algebra permutations

share|cite|improve this question

edited Jul 21 at 20:47

asked Jul 21 at 13:44

UJung

112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I do know the number of elements in Sn
  – UJung
  Jul 21 at 13:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
  – Jyrki Lahtonen
  Jul 21 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 at 13:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See also here.
  – Jyrki Lahtonen
  Jul 21 at 13:54
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  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 at 19:19
  

  
  
  
  

 | 
show 10 more comments

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

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 $iin text1,2,...,n$

For example, for the case n=3, only two pairs ($sigma_1$,$sigma_1$) and ($sigma_2$,$sigma_2$) satisfies all the conditions where
$$sigma_1:=beginpmatrix 1 & 2 & 3 \ 2 & 3 & 1 endpmatrix$$
$$sigma_2:=beginpmatrix 1 & 2 & 3 \ 3 & 1 & 2 endpmatrix$$

abstract-algebra permutations

share|cite|improve this question

edited Jul 21 at 20:47

asked Jul 21 at 13:44

UJung

112

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 $iin text1,2,...,n$

For example, for the case n=3, only two pairs ($sigma_1$,$sigma_1$) and ($sigma_2$,$sigma_2$) satisfies all the conditions where
$$sigma_1:=beginpmatrix 1 & 2 & 3 \ 2 & 3 & 1 endpmatrix$$
$$sigma_2:=beginpmatrix 1 & 2 & 3 \ 3 & 1 & 2 endpmatrix$$

abstract-algebra permutations

share|cite|improve this question

edited Jul 21 at 20:47

asked Jul 21 at 13:44

UJung

112

share|cite|improve this question

share|cite|improve this question

edited Jul 21 at 20:47

edited Jul 21 at 20:47

edited Jul 21 at 20:47

asked Jul 21 at 13:44

UJung

112

asked Jul 21 at 13:44

UJung

112

asked Jul 21 at 13:44

UJung

112

112

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I do know the number of elements in Sn
  – UJung
  Jul 21 at 13:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
  – Jyrki Lahtonen
  Jul 21 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 at 13:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See also here.
  – Jyrki Lahtonen
  Jul 21 at 13:54
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  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 at 19:19
  

  
  
  
  

 | 
show 10 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  I do know the number of elements in Sn
  – UJung
  Jul 21 at 13:47
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
  – Jyrki Lahtonen
  Jul 21 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 at 13:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  See also here.
  – Jyrki Lahtonen
  Jul 21 at 13:54
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  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 at 19:19
  

  
  
  
  

I do know the number of elements in Sn
– UJung
Jul 21 at 13:47

I do know the number of elements in Sn
– UJung
Jul 21 at 13:47

2

2

If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
– Jyrki Lahtonen
Jul 21 at 13:48

If $mu$ and $sigma$ are permutations without fixed points isn't the condition $mu(sigma(i))neqsigma(i)$ automatic? Otherwise $sigma(i)$ would be a fixed point of $mu$, no?
– Jyrki Lahtonen
Jul 21 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 at 13:53

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 at 13:53

See also here.
– Jyrki Lahtonen
Jul 21 at 13:54

See also here.
– Jyrki Lahtonen
Jul 21 at 13:54

1

1

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 at 19:19

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 at 19:19

 | 
show 10 more comments

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