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How many elements of order $n$ does $S_7$ contain? Is there a general formula for to compute how many elements of order $k$ (for a given) in $S_n$?

user26857
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David
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1 Answers1

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You can come up with a formula in the following way. A permutation $\pi$ satisfies $\pi^k = e$ if all cycles have length divisible by $k$. Therefore, using exponential generating functions you can count this number as the coefficient of $x^n/n!$ in $$ \exp\sum_{t|k} \frac{x^t}{t}. $$ In order to compute the number of permutations of order exactly $k$, you can use inclusion-exclusion (in this case, this is also known as Möbius inversion). When $k$ is prime this will be particularly simple.

Yuval Filmus
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  • This is correct except for a minor typographical error which should not be difficult to spot. – Marko Riedel Oct 09 '14 at 21:43
  • Right, thanks. Hopefully I corrected the mistake rather than introducing a new one. – Yuval Filmus Oct 09 '14 at 22:09
  • This was indeed the typo I was refering to. Because there are $\frac{k!}{k} = (k-1)!$ cycles on $k$ labels the EGF is $$\sum_{k\ge 1} \frac{k!}{k} \frac{z^k}{k!} = \sum_{k\ge 1} \frac{z^k}{k}.$$ The EGF you present is a restriction of this one to the set $k|q$ of cycle sizes. (+1). – Marko Riedel Oct 09 '14 at 22:17