This paper presents the number of combinatorially distinct rooted Eulerian planar maps with the number of non-root-vertices and the number of non-root-faces as two parameters. The parametric expressions for determinin...This paper presents the number of combinatorially distinct rooted Eulerian planar maps with the number of non-root-vertices and the number of non-root-faces as two parameters. The parametric expressions for determining the number in tha loopless Eulerian case are also obtained.展开更多
The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the ...The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the special cases including loopless nearly 2-regular maps and simple nearly 2-regular maps in terms of the above three parameters are derived.展开更多
This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, ...This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.展开更多
The functional equation satisfied by the vertex partition function of rooted loopless Eulerianplanar maps is provided. As applications, the enumerating equations for general and regular casesof this kind of maps are a...The functional equation satisfied by the vertex partition function of rooted loopless Eulerianplanar maps is provided. As applications, the enumerating equations for general and regular casesof this kind of maps are also discussed.展开更多
This paper provides some functional equations satisfied by the generatingfunctions for enumerating general rooted planar maps with up to three parameters. Furthermore, thegenerating functions can be obtained explicitl...This paper provides some functional equations satisfied by the generatingfunctions for enumerating general rooted planar maps with up to three parameters. Furthermore, thegenerating functions can be obtained explicitly by employing the Lagrangian inversion. This is alsoan answer to an open problem in 1989.展开更多
This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation ar...This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.展开更多
This paper investigates the number of rooted unicursal planar maps and presents some formulae for such maps with four parameters： the numbers of nonrooted vertices and inner faces and the valencies of two odd vertices.
A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with ...A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters： the number of edges, the number of inner faces and the valencies of the two odd vertices.展开更多
On chromatic enumeration, the first paper which is for rooted planar triangulations was published in 1973. Ten years later, the author generalized the theory into that for rooted nonseparable maps. Very recently, the ...On chromatic enumeration, the first paper which is for rooted planar triangulations was published in 1973. Ten years later, the author generalized the theory into that for rooted nonseparable maps. Very recently, the author also found a展开更多
All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations base...All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations based on the enumerations in [1, 2]. Bender and Richmond gave a survey of the asymptotic enumeration. New asymptotic enumerations here can be considered as a complement and improvement of the survey.展开更多
基金Supported by the Italian National Research Councilthe National Natural Science Foundation of China.
文摘This paper presents the number of combinatorially distinct rooted Eulerian planar maps with the number of non-root-vertices and the number of non-root-faces as two parameters. The parametric expressions for determining the number in tha loopless Eulerian case are also obtained.
文摘The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the special cases including loopless nearly 2-regular maps and simple nearly 2-regular maps in terms of the above three parameters are derived.
基金This research was partially supported by the U. S. National Science Foundation under Grant Number ECS 85-03212 and by the National Natural Science Foundation of China as well. And, it was completed during the author's stay at RUTCOR, The State Univerity
文摘This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.
基金This project is supported partially by the National Natural Science Foundation of China Grant 18971061
文摘The functional equation satisfied by the vertex partition function of rooted loopless Eulerianplanar maps is provided. As applications, the enumerating equations for general and regular casesof this kind of maps are also discussed.
基金Project 10271017 supported by National Natural Science Foundation of China
文摘This paper provides some functional equations satisfied by the generatingfunctions for enumerating general rooted planar maps with up to three parameters. Furthermore, thegenerating functions can be obtained explicitly by employing the Lagrangian inversion. This is alsoan answer to an open problem in 1989.
文摘This paper is concerned with the existence of analytic invariant curves for a planar mapping. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. In this paper, we discuss not only the general case, but also the critical cases as well, in particular, the case where β is a unit root is discussed.
基金Supported by the National Natural Science Foundation of China(No.10271017)the Natural Science Foundation Project of Chongqing(N0.cstc2012jjA00041)Chongqing Innovation Fund(grant no.KJTD201321)
文摘This paper investigates the number of rooted unicursal planar maps and presents some formulae for such maps with four parameters： the numbers of nonrooted vertices and inner faces and the valencies of two odd vertices.
基金Supported by the National Natural Science Foundation of China(No.10271017,11371133,11571044)the Natural Science Foundation Project of Chongqing(No.cstc2012jj A00041,cstc2014jcyj A00041)the Innovation Foundation of Chongqing(No.KJTD201321)
文摘A map is 4-regular unicursal if all its vertices are 4-valent except two odd-valent vertices. This paper investigates the number of rooted 4-regular unicursal planar maps and presents some formulae for such maps with four parameters： the number of edges, the number of inner faces and the valencies of the two odd vertices.
基金This research was supported by the National Natural Science Foundation of China.
文摘On chromatic enumeration, the first paper which is for rooted planar triangulations was published in 1973. Ten years later, the author generalized the theory into that for rooted nonseparable maps. Very recently, the author also found a
基金Project supported by the National Natural Science Foundation of China.
文摘All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations based on the enumerations in [1, 2]. Bender and Richmond gave a survey of the asymptotic enumeration. New asymptotic enumerations here can be considered as a complement and improvement of the survey.
