  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 Jackson's 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 Jackson's 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.
