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

Bijective Enumeration of 3-Factorizations of an N-Cycle