2D Toda t functions, weighted Hurwitz numbers and the Cayley graph: Determinant representation and recursion formula

We generalize the determinant representation of the Kadomtsev-Petviashvili tau functions to the case of the 2D Toda tau functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda tau functions, for which we give a determinant representation of weighted Hurwitz numbers. Then, we can get a finite-dimensional equation system for the weighted Hurwitz numbers H-G(d)(sigma, omega) with the same dimension |sigma| = |omega| = n. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension 0, 1, 2, 3 and give a recursion formula for calculating the higher dimensional weighted Hurwitz numbers. Finally, we get a matrix representation for the Hurwitz numbers and obtain a determinant representation of weighted paths in the Cayley graph. Published under an exclusive license by AIP Publishing.