Long Cycle Factorizations : Bijective Computation in the General Case

TL;DR

This paper presents a combinatorial approach to compute the number of ordered factorizations of long cycles in symmetric groups, providing a new bijective method and a purely combinatorial formula for their generating series.

Contribution

It introduces a bijective technique to derive a combinatorial formula for long cycle factorizations, improving upon previous character-based methods.

Findings

Derived a new combinatorial formula for factorizations

Provided a bijective proof for the generating series

First purely combinatorial evaluation of these series

Abstract

This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series.