Hurwitz numbers, ribbon graphs, and tropicalization

TL;DR

This paper explores the multiple equivalent definitions of double Hurwitz numbers, highlighting their relationships through combinatorial algorithms and graph theory, and discusses their piecewise polynomial nature.

Contribution

It provides a simple combinatorial algorithm to convert between permutation and ribbon graph definitions of double Hurwitz numbers, and compares proofs of their piecewise polynomiality.

Findings

Established equivalences between different definitions of double Hurwitz numbers

Developed a direct combinatorial algorithm for conversion between definitions

Compared proofs of piecewise polynomiality of double Hurwitz numbers

Abstract

Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count 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, double Hurwitz numbers have been expressed as 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 double Hurwitz numbers are piecewise polynomial. We use our algorithm to compare these two proofs.