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Math Help - Permutations

  1. #1
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    poland
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    Permutations

    1. How many cyclic permutations are there of set with n elements
    2. Permutation  \pi e S_{n} has k cycles of powers c_{1}..c_{k}
    show that: a) sgn(\pi)=\prod^{k}_{i=1} (-1)^{c_{i}-1}
    b) sgn(\pi)=(-1)^{n-k}

    for 1.. isn't it just simply n! if the cyclic permutation is defined as one with just 1 cycle and \sum_{i=2}^{n} i! if it's defined as one with just one and only cycle that has more than 2 elements in it? how to prove it if it's true?

    2. No idea, also power of cycle tells us how many elements each cycle has, right?
    Last edited by aquance; April 5th 2014 at 06:16 AM.
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  2. #2
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    Apr 2014
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    poland
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    Re: Permutations

    OK so for 1 I know it's (n-1)! (because cycle 1234 is the same as 2341) but I don't know how to prove it

    for 2a) we know that if cycle has ck elements then it's sign is (-1)ck-1 and we know that sgn(a*b)=sgn(a)*sgn(b). Does the second equality mean that  sgn(c_{1}* ... * c_{k}) = \prod_{i=1}^{k} sgn(c_{i}) ? If no, how to prove it?


    for 2b) we would have to prove that \sum_{i=1}^{k} c_{i} = n but I don't know how to prove it - help?
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