本文研究了四种类型的带根可平面地图:单行可平面图、4-正则单行可平面地图、不可分单行可平面地图及广义冬梅地图的计数问题. 全文共分为四章. 第一章讨论带根单行可平面地图的计数,提供了以其边数(或非根点数、内面数)和两个奇点次为参数的计数函数所满足的计数方程,并导出了它们的所有显式,其中有两个是无和的. 第二章对以非根点数(或边数、内面数)和两个奇点次为参数的带根4-正则单行可平面地图进行了计数, 并且得到了几个无和式. 第三章研究带根不可分单行可平面地图的计数, 给出了节点剖分方程及以非根点数、内面数和两个奇点次为四参数的计数函数所满足的一些函数方程. 第四章对以根点次、非根点数和内面数为三参数的广义冬梅地图进行了计数,并且解决了两个三次方程.

The author investigates the enumerative problems of four types of rooted planar maps:unicursal planar maps, 4-regular unicursal planar maps, nonseparable unicursal planar maps and generalized wintersweets. The thesis is divided into four chapters.The first chapter discusses the enumeration of rooted unicursal planar maps, provides some functional equations satisfied by the enumerating functions with edge number (or the numbers of nonrooted vertices, inner faces) and the valencies of two odd vertices of the maps as parameters. All explicit expressions of them are derived and two of them are summation-free.In the second chapter rooted 4-regular unicursal planar maps are counted with formulae with respect to parameter:the number of nonrooted vertices (or the numbers of edges, inner faces) and the valencies of two odd vertices. Several sum-free formulae are obtained.The third chapter researches the enumeration of rooted nonseparable unicursal planar maps, presents the vertex partition equation and some functional equations satisfied by the enumerating functions with the numbers of nonrooted vertices, inner faces and the valencies of two odd vertices of the maps as four parameters.In the fourth chapter generalized wintersweets are counted with formulae with respect to three parameters:the root-vertex valency, the numbers of nonrooted vertices and inner faces. Two cubic equations are solved.