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2D Toda $τ$ Functions, Weighted Hurwitz Numbers and the Cayley Graph: Determinant Representation and Recursion Formula

We generalize the determinant representation of the KP $τ$ functions to the case of the 2D Toda $τ$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $τ$ 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 $H^d_{G}(σ,ω)$ with the same dimension $|σ|=|ω|=n$. Using this equation system, we calculated the value of the weighted Hurwitz numbers with dimension $0,\,1,\,2$ and give a recursion formula to calculating the higher dimensional weighted Hurwitz numbers. For any given weighted generating function $G(z)$, the weighted Hurwitz number degenerates into the Hurwitz numbers when $d=0$. We get a matrix representation for the Hurwitz numbers. The generating functions of weighted paths in the Cayley graph of the symmetric group are a parametric family of 2D Toda $τ$ functions; for which we obtain a determinant representation of weighted paths in the Cayley graph.