Permutations with pl-th roots

AbstractA permutation of n objects is of cycle type (j1,…,jn) if it has jk, k=1,…,n, cycles of length k in its disjoint cycle decomposition. Let Sn be the symmetric group of degree n, and d,s∈Z+, the set of positive integers. We enumerate the permutations in Sn of cycle type (j1,…,jn), where jk ≡ 0 mod s if k ≡ 0 mod d.For the case d=p, a prime number, and s=pl, l∈Z+, this gives the number of permutations in Sn with pl-th roots