Bijections and symmetries for the factorizations of the long cycle

We study the factorizations of the permutation (1,2,…,n) into k factors of given cycle types. Using representation theory, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases k=2,3 Schaeffer and Vassilieva gave a combinatorial proof of Jacksonʼs formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of (1,2,…,n) into k factors for all k. We thereby obtain refinements of Jacksonʼs formulas which extend the cases k=2,3 treated by Morales and Vassilieva. Our bijections are described in terms of “constellations”, which are graphs embedded in surfaces encoding the transitive factorizations of permutations.