Rooted maps on orientable surfaces, Riccati's equation and continued fractions
Rooted maps on orientable surfaces, Riccati's equation and continued fractions
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DOI：
10.1016/s0012-365x(99)00197-1
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发表时间：
2000-03-28
影响因子：
0.8
通讯作者：
Béraud, JF
中科院分区：
文献类型：
--
作者：
Arquès, D;Béraud, JF
本文提出了一种不考虑亏格的有根映射的新方法。我们证明了存在一种新的类型的方程的生成系列的这些地图枚举的边和顶点。这是黎卡提方程。这似乎是第一次，这样一个微分方程出现在枚举的根源地图。求解这个方程导致不同的封闭形式的研究生成系列。最有趣的后果是一个非常好的连续分数导致一个新的方程推广著名的戴克方程有根平面树的生成函数的发展。在第二部分中，我们也得到了一个微分方程的生成系列的根树，无论属，关于边缘。这也导致了一个连续的分数生成系列的根属独立的树木和一个意想不到的关系之前生成系列的树和根地图。(C)2000 Elsevier Science B.V.保留所有权利。
We present a new approach in the study of rooted maps without regard to genus. We prove the existence of a new type of equation for the generating series of these maps enumerated with respect to edges and vertices. This is Riccati's equation. It seems to be the first time that such a differential equation appears in the enumeration of rooted maps. Solving this equation leads to different closed forms of the studied generating series. The most interesting consequence is a development of this generating function in a very nice continued fraction leading to a new equation generalizing the well-known Dyck equation for rooted planar trees. In a second part, we also obtain a differential equation for the generating series of rooted trees regardless of the genus, with respect to edges. This also leads to a continued fraction for the generating series of rooted genus independent trees and to an unexpected relation between both previous generating series of trees and rooted maps. (C) 2000 Elsevier Science B.V. All rights reserved.