Bijective Enumeration of 3-Factorizations of an N-Cycle

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