Double Hurwitz numbers via the infinite wedge

We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula in the work of Goulden, Jackson, and Vakil for one part double Hurwitz numbers. Consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Strong Piecewise Polynomiality Conjecture of their work.

The proof is an application of Okounkov's expression for double Hurwitz numbers in terms of operators on the infinite wedge. We begin with a introduction to the infinite wedge tailored to our use.