On the number of mth roots of permutations

Let m be a fixed positive integer. It is well-known that a permutation $σ$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of σ. Let p_m(n) be the probability that a random n-permutation has an mth root. We also include a proof that p_m(jq)=p_m(jq+1)=... =p_m(jq+(q-1)) where j=0,1,... and m is a power of prime q.

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