Classical Hurwitz numbers and related combinatorics

We give a polynomial-time algorithm of computing the classical Hurwitz numbers $H_{g,d}$, which were defined by Hurwitz 125 years ago. We show that the generating series of $H_{g,d}$ for any fixed $g\geq2$ lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the intersection theory on moduli spaces of algebraic curves, such that their generating series belong to the same Lambert ring. Several asymptotics of these numbers (for large $g$ or for large $d$) are obtained.