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

TL;DR

This paper explores the multiplication of cycles in the symmetric group, generalizing a theorem about permutations, and provides formulas for cycle distributions and separation probabilities, advancing understanding of cycle interactions.

Contribution

It generalizes Boccara's theorem on cycle products, introduces new approaches including inductive proofs, and derives formulas for cycle distributions and separation probabilities.

Findings

Number of ways to write an odd permutation as an n-cycle and (n-1)-cycle is permutation-independent.

Provides a formula for the distribution of cycle counts in fixed-length cycle products.

Connects cycle multiplication results to separation probabilities in permutations.

Abstract

We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols as a product of an $n$-cycle and an $n-1$-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, Du, Morales and Stanley (2014).