论文标题
阳性属中双变量生成一系列地图的合理性的组合证明
Combinatorial proof for the rationality of the bivariate generating series of maps in positive genus
论文作者
论文摘要
在本文中，我们给出了第一个组合证明，以证明由顶点和面部列举的正属中生成一系列地图的合理性方案，这是Bender，Canfield和Richmond在1993年首次通过纯计算技术获得的。为此，我们依靠第二作者在这些地图和装饰单细胞图的家族之间获得的第二作者获得的两者。我们的主要贡献包括对这个地图家族的精细分析。作为副产品，我们还获得了一个新的，更简单的组合证明，以列举其边缘数量的生成一系列地图的合理性方案，最初是由Bender和Canfield在1991年在计算中获得的，并在2019年由第二作者在计算中进行了计算。
In this paper, we give the first combinatorial proof of a rationality scheme for the generating series of maps in positive genus enumerated by both vertices and faces, which was first obtained by Bender, Canfield and Richmond in 1993 by purely computational techniques. To do so, we rely on a bijection obtained by the second author in a previous work between those maps and a family of decorated unicellular maps. Our main contribution consists in a fine analysis of this family of maps. As a byproduct, we also obtain a new and simpler combinatorial proof of the rationality scheme for the generating series of maps enumerated by their number of edges, originally obtained computationally by Bender and Canfield in 1991 and combinatorially by the second author in 2019.
