Enumeration of almost polynomial rational functions with given critical values

Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by Hurwitz [A. Hurwitz, Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Mathematische Annalen 39 (1891) 1–61. [7]]. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instance, those realized by polynomials in one complex variable. In this paper we obtain an explicit answer for a large new family of coverings, namely, the coverings realized by simple almost polynomials, defined below. Unlike most other results in the field, our formula is obtained by elementary methods.