### Generating functions for weighted Hurwitz numbers *[Cross-Listing]*

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Weighted Hurwitz numbers for $n$-sheeted branched coverings of the Riemann sphere are introduced, together with associated weighted paths in the Cayley graph of $S_n$ generated by transpositions. The generating functions for these, which include all formerly studied cases, are 2D Toda $\tau$-functions of generalized hypergeometric type. Two new types of weightings are defined by coefficients in the Taylor expansion of the exponentiated quantum dilogarithm function. These are shown to provide $q$-deformations of strictly monotonic and weakly monotonic path enumeration generating functions. The standard double Hurwitz numbers are recovered from both types in the classical limit. By suitable interpretation of the parameter $q$, the corresponding statistical mechanics of random branched covers is related to that of Bose gases.