Minimal Transitive Factorizations of Permutations into Cycles

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings, that is, we study the number Hα(i2, i3, . . . ) of ways a given permutation (with cycles described by the partitionα) can be decomposed into a product of exactly i2 2-cycles, i3 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 determine Hα(i2, i3, . . . ) when α has 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 us to generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of their analysis. Department of Mathematics and Computing Science, Saint Mary’s University, Halifax, NS, B3H 3C3 e-mail: [email protected] Received by the editors October 20, 2006. AMS subject classification: Primary: 05A15; secondary: 05E10. c ©Canadian Mathematical Society 2009. 1092