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How many ordered pairs of permutations $(\pi , \sigma )$ in $S_n$ such that $\pi \circ \sigma =\sigma \circ \pi $.

I think i need consider group acts on itself by conjugation $\pi (\sigma )=\pi \circ \sigma \circ \pi ^{-1}$. I need help or hint.

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Hints: (1) the number of permutations commuting with a given one depend only on the conjugacy class of the latter (2) for any finite group acting on a finite set (here you can take $S_n$ acting on itself by conjugation) the number of orbits equals the average number (over the group) of fixed points. Come to think of it, just hint (2) should do the trick.

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  • $\begingroup$ I solved it. Thank you. $\endgroup$ – JJMM Apr 13 '15 at 15:36

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