  We introduce a new approach to an enumerative problem closely linked with the
geometry of branched coverings; that is, we study the number of ways a
permutation can be decomposed into a product of a given number of 2-cycles,
3-cycles, etc. with certain minimality and transitivity conditions imposed on
the factors. The method is to encode such factorizations as planar maps with
certain "descent structure" and apply a new combinatorial decomposition to make
their enumeration more manageable. We apply our technique to count
factorizations of permutations with one or two parts, extending earlier work of
Goulden and Jackson. We also show how these methods are readily modified to
count inequivalent factorizations, where equivalence is defined by permitting
commutations of adjacent disjoint factors. Our technique permits a substantial
generalization of recent work of Goulden, Jackson, and Latour, while allowing
for a considerable simplification of their analysis.
