Two enumerative results on cycles of permutations

• Authors: Richard P. Stanley
• Venue: European Journal of Combinatorics
• Year: 2011

Answering a question of Bona, 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 12-2(n-1)(n+2) if n is even. Another result concerns the polynomial P"@l(q)=@?"wq^@k^(^(^1^,^2^,^...^,^n^)^@?^w^), where w ranges over all permutations in the symmetric group S"n of cycle type @l, (1,2,...,n) denotes the n-cycle 1-2-...-n-1, and @k(v) denotes the number of cycles of the permutation v. A formula is obtained for P"@l(q) from which it is deduced that all zeros of P"@l(q) have real part 0.