Two Enumerative Results on Cycles of Permutations

TL;DR

This paper investigates cycle probabilities in permutation products and derives formulas for generating functions, revealing properties of their zeros, advancing understanding of permutation cycle structures.

Contribution

It provides new enumerative results on cycle probabilities and generating functions for permutation products, answering a specific open question and analyzing zeros of these functions.

Findings

Probability that 1 and 2 are in the same cycle is 1/2 for odd n.

Derived a formula for the generating function P_h(q).

Zeros of P_h(q) have real part 0.

Abstract

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.