Rooted maps on orientable surfaces, Riccati's equation and continued fractions
type: Item
instance of: scholarly article
title: Rooted maps on orientable surfaces, Riccati's equation and continued fractions
zbMATH Open document ID: 0944.05056
DOI: 10.1016/S0012-365X(99)00197-1
author: Jean-François Béraud
author: Didier G. Arquès
published in: Discrete Mathematics
publication date: 2000-09-15T00:00:00Z
review text: 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.
reviewed by: Yanpei Liu
Mathematics Subject Classification ID: 05C30
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 1439819
zbMATH Keywords: map
zbMATH Keywords: orientable surface
zbMATH Keywords: enumeration
zbMATH Keywords: Riccati's equation
zbMATH Keywords: continued fraction
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1439819