L Moser and M Wyman, *On solutions of $x^d = 1$ in symmetric groups*, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations:
$$f\_d(n):=\#\{\pi\in\mathfrak{S}\_n:\, \pi^d=1\}.$$
In particular, $f\_2(n)$ counts the number of *involutions* in the symmetric group $\mathfrak{S}\_n$. 



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> **Question 1.** I recall (but forgot where) $f\_d(n)$ enumerate *pseudoknots* in RNA molecular folding. Can you explain this or point me to a readable reference?
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Also,



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> **Question 2.** Does this generating function hold true?
>  $$\sum\_{n\geq0}f\_d(n)\frac{x^n}{n!}=e^{\sum\_{c\,\vert\,d}\frac1cx^c}.$$
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Solutions of $x^d=1$ in the symmetric group
Q2. Of course, this is a general thing for exponential generating functions. Assume that $a(n)$ is the number of ways to make lunch from $n$ distinct ingredients, $f(z)=\sum \frac{a(n)}{n!} z^n$ is an exponential generating function. Then, say, $f^2$ is an exponential generating function for making two enumerated lunches from $n$ ingredients, $f^2/2$ for two not enumerated lunches, $f^3/3!$ for three non-enumerated lunches and so on. Totally $\exp(f)$ for arbitrary number of lunches. In our situation lunch is a cyclic ordering of ingredients when the number of ingredients divides $d$.


