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Sep 23, 2015 at 2:45 comment added user61522 I was trying to avoid using the classification of irreps of $S_n$.
Sep 23, 2015 at 2:24 comment added darij grinberg If you really want elementary, you cannot beat the proof via the Young-tableau description of the irreps.
Sep 23, 2015 at 1:59 history answered user61522 CC BY-SA 3.0