Realizability of branched coverings of

The paper is concerned with the realizability problem of branched coverings of S 2 , namely, given a collection of partitions of a positive integer d, whether there is a branched covering M → S 2 with it as branch data. By reducing to the enumeration of a special type of fat graphs, we deduce a formula in terms of characters of symmetric groups to decide the problem. Computation further shows that the difficulty of the problem only lies in the case when the genus of the covering surface is 0 or 1.