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Working paper

On the number of coverings of the sphere ramified over given points

We present the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus g compact oriented surface not ramified outside of a given set of m+1 points in the target, fixed ramification type over one point, and arbitrary ramification types over the remaining m points. We present the genus expansion of this generating function and prove, that the generating function of coverings of genus 0 satisfies some system of differential equations.