Enumeration of Planar Constellations

The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n ≥ 1, and let σ₀ be a permutation of _n having d_i cycles of length i, for i ≥ 1. Let m ≥ 2. We prove that the number of m-tuples (σ₁,…,σ_m) of permutation of _n such that

• σσ···σ = σ,

• the group generated by σ,…,σ acts transitively on {1, 2,…,},

• ∑ (σ) = ( − 1) + 2, where (σ) denotes the number of cycles of σ

A one-to-one correspondence relates these m-tuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m = 2, we recover a formula of Tutte for the number of Eulerian maps. The proof relies on the idea that maps are conjugacy classes of trees and extends the method previously applied to Eulerian maps by the second author. Our result might remind the reader of an old theorem of Hurwitz, giving the number of m-tuples of transpositions satisfying the above conditions. Indeed, we show that our result implies Hurwitz' theorem. We also briefly discuss its implications for the enumeration of nonequivalent coverings of the sphere.