Two enumerative results on cycles of permutations

Answering a question of Bóna, it is shown that for n ≥ 2 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1, 2, . . . , n} is 1/2 if n is odd and 1 2 − 2 (n−1)(n+2) if n is even. Another result concerns the polynomial Pλ(q) = ∑ w q κ((1,2,...,n)·w), where w ranges over all permutations in the symmetric group Sn of cycle type λ, (1, 2, . . . , n) denotes the n-cycle 1 → 2 → · · · → n → 1, and κ(v) denotes the number of cycles of the permutation v. A formula is obtained for Pλ(q) from which it is deduced that all zeros of Pλ(q) have real part 0.