Hurwitz numbers, ribbon graphs, and tropicalization

The double Hurwitz number Hg(µ, ν) has at least four equivalent definitions. Most naturally, it counts the covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain collections of permutations. More recently, it has been shown to be equivalent to a count of certain ribbon graphs, or as a weighted count of certain labeled graphs. This note is an expository account of the equivalences between these definitions, with a few novelties. In particular, we give a simple combinatorial algorithm to pass directly between the permutation and ribbon graph definitions. The two graph theoretic points of view have been used to give proofs that Hg(µ, ν) is piecewise polynomial in the µi and νj . We use our algorithm to compare these two proofs.