Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter b with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate b-weighting that “measures” in some sense their non-orientability.