The University of Michigan Combinatorics Seminar
Double Hurwitz numbers in genus zero count the covers of CP1 by CP1 which have two non-simple branching points with a given branching behaviour. By the recent result of Goulden, Jackson, and Vakil, these numbers are given by piecewise polynomial functions in the multiplicities of the preimages of the branching points. We describe a partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of the polynomials associated with two adjacent chambers. We also provide an explicit formula for the polynomial associated with a particular distinguished "totally nonnegative" chamber. These results enable us to explicitly calculate many double Hurwitz numbers.
This is joint work with S.Shadrin and A.Vainshtein.