Two enumerative results on cycles of permutations

Answering a question of Bóna, it is shown that for n≥2n≥2 the probability that 1 and 2 are in the same cycle of a product of two nn-cycles on the set {1,2,…,n}{1,2,…,n} is 1/21/2 if nn is odd and 12−2(n−1)(n+2) if nn is even. Another result concerns the polynomial Pλ(q)=∑wqκ((1,2,…,n)⋅w)Pλ(q)=∑wqκ((1,2,…,n)⋅w), where ww ranges over all permutations in the symmetric group SnSn of cycle type λλ, (1,2,…,n)(1,2,…,n) denotes the nn-cycle 1→2→⋯→n→11→2→⋯→n→1, and κ(v)κ(v) denotes the number of cycles of the permutation vv. A formula is obtained for Pλ(q)Pλ(q) from which it is deduced that all zeros of Pλ(q)Pλ(q) have real part 0.