On products of long cycles: short cycle dependence and separation probabilities

Abstract

We present various results on multiplying cycles in the symmetric group. One result is a generalisation of the following theorem of Boccara (Discret Math 29:105–134, 1980): the number of ways of writing an odd permutation in the symmetric group on $\mathit{n}$ symbols as a product of an $\mathit{n}$-cycle and an ($\mathit{n}$−1)-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths. Another application is related to the recent notion of separation probabilities for permutations introduced by Bernardi et al. (Comb Probab Comput 23:201–222, 2014).