Rooted maps on orientable surfaces, Riccati's equation and continued fractions
This article provides a functional equation for enumerating rooted maps on orientable surfaces without considering the genus. (The case of edges appears in the reviewer's book [Enumerative theory of maps (Kluwer, 1999), page 268, eq. 8.5.10] as well.) This is the type of Riccati's equation. By employing continued fractions, a solution of the equation is expressed. Although Euler got a solution to Riccati's equation by this approach in the 18th century, this method is firstly appearing in enumerating maps to the reviewer's knowledge. In consequence, an explicit formula for the number of rooted maps on all orientable surfaces with the size as the parameter is found as a multisummation with alternative terms. However, trees appearing here should be replaced by orientable maps with only one vertex.
scientific article; zbMATH DE number 1439819