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Let $c \binom n k$ denote the number of permutations in $S_n$ with $k$ cycles.

Find formulas for $c \binom n {n-2}$ and for $c \binom n 2$, and double-check that they hold for $n = 4$.

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What have you tried? What tools are you to use? –  vonbrand Feb 13 '13 at 11:09
    
oeis.org/A008275 –  UwF Feb 13 '13 at 13:42
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Hint: both of these are susceptible to direct computation. For the first, if $n$ is large you will have to have many cycles with just one element. If you think about that, there aren't many patterns to break the $n$ elements among $n-2$ cycles-how many? Then count the ways to assign the $n$ elements. For the second, you have to break $n$ into two summands-how many ways to do that? Again, how many ways to assign the elements. What do you need to be careful of if $n$ is even?

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