Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

TL;DR

This paper introduces mixed double Hurwitz numbers, showing they satisfy the 2-Toda hierarchy and are piecewise polynomial, thus unifying and extending previous results in algebraic combinatorics and integrable systems.

Contribution

It generalizes classical and monotone double Hurwitz numbers, proving their generating series solves the 2-Toda hierarchy and establishing their piecewise polynomiality.

Findings

Generating series solves the 2-Toda hierarchy.

Mixed double Hurwitz numbers are piecewise polynomial.

Unification of classical and monotone Hurwitz numbers.

Abstract

This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.