Given $\mathfrak S_n$, I am trying to find the number of distinct permutations $y$ such that a permutation $x$ exists with $x^r = y$. Here power means composition.
The paper https://core.ac.uk/download/pdf/82287751.pdf gives an estimate for $r=2$. I am trying to calculate such bounds for $r=3$ and $6$.
Kindly provide some references on how to proceed.