This question was moved from the main discussion page to here since no one has answered it after a week:
I'm a little confused about cycle notation. I'm trying to find the permutation described by an example in the book. It says that a permutation (13)(27)(456)(8)(1237)(648)(5) can be expressed in disjoint cycle form as (1732)(48)(56). I believe the resulting permutation should be found as follows:
[1 2 3 4 5 6 7 8]
Locations 5 and 6 swap. Locations 4 and 8 swap. Location 1 becomes 7, 7 becomes 3, 3 becomes 2, 2 becomes 1.
So the resulting permutation is [7 1 2 8 6 5 3 4].
However, when I try to compute this using the original redundant form, I get something different:
[1 2 3 4 5 6 7 8]
(648): Location 4 goes to location 6. 8 to 4. 6 to 8.
[1 2 3 8 5 4 7 6]
(1237): 2 to 1. 3 to 2. 7 to 3. 1 to 7.
[2 3 7 8 5 4 1 6]
(456): 5 to 4. 6 to 5. 4 to 6.
[2 3 7 5 4 8 1 6]
(27): Swap locations 2 and 7.
[2 1 7 5 4 8 3 7]
(13): Swap locations 1 and 3.
[7 1 2 5 4 8 3 6]
This time I get a different permutation. It would appear I'm doing this all wrong. Can anyone help?
If you have something in cycle notation then the meaning of a cycle such as (648) is that the content of box 6 moves to box 4, that of box 4 to box 8 and that of box 8 to box 6. If you apply the permutation (13)(27)(456)(8)(1237)(648)(5) to [12345678] one by one starting from the right you get
[1 2 3 4 5 6 7 8] -> [1 2 3 4 5 6 7 8] -> [1 2 3 6 5 8 7 4] -> [7 1 2 6 5 8 3 4] -> [7 1 2 6 5 8 3 4] -> [7 1 2 8 6 5 3 4] -> [7 3 2 8 6 5 1 4] -> [2 3 7 8 6 5 1 4].
You can directly determine this by tracing the 8 numbers through the 7 cycles above. For example, the location of 1 is 1 -> 1 -> 1 -> 2 -> 2 -> 2 -> 7 -> 7. (It is affected by the 1237-cycle the fist time, when it moves to position 2, and the second time by the 27-cycle when it moves to position 7.)
Similarly, 7 is located in positions 7 -> 7 -> 7 -> 1 -> 1 -> 1 -> 1 -> 3. (It is moved the first time because of the 1237-cycle after which it's in position 1, and the second time by the 13-cycle that moves it to position 3.)
Thanks for the explanation. I guess I was doing it backwards.
