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Thread: permutations

  1. #1
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    permutations

    Waazup !

    I have tha following permutations:

    1) (124)(365)

    2)
    (23)(41)(56)

    3)
    (35)(561)

    4) (13)(254)(6)

    5) (1)(254)(36)

    6) id

    I've used the rule that, if the cycle-length is even then the permutation is odd and if the cycle-length is odd then the permutation is even.
    So 1) is odd, 2) is even, 3) is odd, 4) is even, 5) is even and 6) is odd.

    What ya'll think?

    Get back 2 me ...
    peace out
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  2. #2
    MHF Contributor

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    Re: permutations

    That makes me wonder if you know the definition of "even" and "odd" permutation.
    (356) maps 3 to 5, 5 to 6 and 6 to 3: (123456) is mapped to (125463). Then (124) map 1 to 2, 2 to 4, and 4 to 1: (125463) is mapped to (245163).
    Putting the two together, (123456) is mapped to (245163). We could also do that by swapping 1 and 2 to get (213456) swapping 3 and 4: (214356), swapping 1 and 4: (241356) swapping 3 and 5: (241536), swapping 3 and 6: (241563), and, finally, swapping 1 and 5: (245163). That took 5 "swaps" of two numbers so it is an "odd" permutation.

    (23)(41)(56) consists of three independent "swaps". It is an odd permutation, not even.

    (561) maps 5 to 6, 6 to 1, and 1 to 5: (123456) is mapped to (523461).
    We could do that (this is the hard way) by: swap 1 and 2: (213456), swap 1 and 3: (231456), swap 1 and 4: (234156), swap 1 and 5, (234516), swap 1 and 6: (234561), swap 4 and 5235461), swap 3 and 5: (253461), finally swap 2 and 5: (253461). That required 8 swaps so this is an even permutation. We could also do that by using swaps that are not adjacent: swap 1 and 6: (623451), swap 6 and 5: (623461). That requires only 2 swaps instead of 8, but it is still an even number of swaps of two numbers so this is an even permutation , not odd. "Odd" and "even" permutations is not just the number of digits shown.
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