Macdonald-Hurwitz numbers and alternating path operators

Hurwitz numbers count inequivalent branched coverings of the sphere by an orientable surface with a given number of ramification points. Some specializations of the generating function of Hurwitz numbers are known to be functions of the KP and the 2-Toda hierarchy. Using Macdonald polynomials, we introduce a q,t deformation of the generating function of Hurwitz numbers, and we prove that it is characterized by a family of differential equations. The key tool of the proof is a new family of operators encoded by alternating paths.

This talk is based on a joint work with Valentin Bonzom and Maciej Dołęga.

For information about the Richard P. Stanley Seminar in Combinatorics, visit https://math.mit.edu/combin/