TITLE: Solutions of $x^d=1$ in the symmetric group
QUESTION [9 upvotes]: 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 $\mathfrak{S}_n$. 

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?

Also,

Question 2. Does this generating function hold true?
  $$\sum_{n\geq0}f_d(n)\frac{x^n}{n!}=e^{\sum_{c\,\vert\,d}\frac1cx^c}.$$

REPLY [4 votes]: For Q1, there is a quite nicely written 2014 preprint by Cheda and Gupta.