Two Enumerative Results on Cycles of Permutations

Details Two Enumerative Results on Cycles of Permutations Two Enumerative Results on Cycles of Permutations

2009-01-14

arxiv.org/abs/0901.2008v1

Mathematics

Combinatorics

10 pages

Scientific paper

Answering a question of Bona, it is shown that for n>1 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 generating function P_h(q) for the number of cycles of the product (1,2,...,n)w, where w ranges over all permutations of 1,2,...,n of cycle type h. A formula is obtained for P_h(q) from which it is proved that the zeros of P_h(q) have real part 0.

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