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 P λ ( q ) = ∑ w q κ ( ( 1 , 2 , … , n ) ⋅ w ) , where w ranges over all permutations in the symmetric group S n 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 P λ ( q ) from which it is deduced that all zeros of P λ ( q ) have real part 0.