Bijective Enumeration of 3-Factorizations of an N-Cycle

TL;DR

This paper introduces a novel combinatorial method to count factorizations of a long cycle into three permutations within the symmetric group, providing an elegant and explicit formula.

Contribution

It presents the first purely combinatorial bijection for counting 3-factorizations of an N-cycle, advancing understanding of symmetric group factorizations.

Findings

Derived an explicit formula for 3-factorizations of an N-cycle

Established a new bijection for partitioned 3-cacti

Provided the first combinatorial proof of this enumeration

Abstract

This paper is dedicated to the factorizations of the symmetric group. Introducing a new bijection for partitioned 3-cacti, we derive an el- egant formula for the number of factorizations of a long cycle into a product of three permutations. As the most salient aspect, our construction provides the first purely combinatorial computation of this number.