Question about symmetric group Watch

gangsta316
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Report Thread starter 7 years ago
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How many elements of cycle shape (1, 1, 2, 2) are there in S6? I got 6C2 * 4C2 but the answer is supposed to be half of this. Here's my argument: choose 2 of these to get mapped to themselves (6C2), then out of the 4 remaining, choose 2 of them to go into the first 2 cycle blob (so 4C2; we only need to choose 2: order doesn't matter because the cycle (i j) is the same as the cycle (j i)), and then the entries of the final 2 cycle are chosen for you. Now I know that we have to be careful not to double count here, because if we re-order a 2 cycle, we have the same 2, cycle. But where have I gone wrong?
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nuodai
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I think your error is in not taking into account that if \sigma, \tau are disjoint cycles then \sigma \tau = \tau \sigma, so you're double-counting, say, (1\ 2)(3\ 4) and (3\ 4)(1\ 2), both of which are the same element in S_6.
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gangsta316
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Report Thread starter 7 years ago
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(Original post by nuodai)
I think your error is in not taking into account that if \sigma, \tau are disjoint cycles then \sigma \tau = \tau \sigma, so you're double-counting, say, (1\ 2)(3\ 4) and (3\ 4)(1\ 2), both of which are the same element in S_6.
Ugh. Of course! Thank you.
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