Enumerative Theory of Maps（英文影印版）（硬皮精装）

　　this monograph provides a unified theory of maps and their enumerations. the crucial idea is to suitably decompose the given set of maps for extracting a functional equation, in order to have advantages for solving or transforming it into those that can be employed to derive as simple a formula as possible. it is shown that the foundation of the theory is for rooted planar maps, since other kinds of maps including nonrooted (or symmetrical ) ones and those on general surfaces have been found to have relationships with particular types in planar cases. a number of functional equations and close formulae are discovered in an exact or asymptotic manner. .
　　 this book will be of interest to college teachers, graduate students working in mathematics, especially in combinatorics and graph theory, functional and approximate analysis and algebraic systems. ...

preface.
chapter 1 preliminaries
§1.1 maps
§1.2 polynomials on maps
§1.3 enufunctions
§1.4 polysum functions
§1.5 the lagrangian inversion
§1.6 the shadow functional
§1.7 asymptotic estimation
§1.8 notes
chapter 2 outerplanar maps
§2.1 plane trees
§2.2 wintersweets
§2.3 unicyclic maps
§2.4 general outerplanar maps
§2.5 notes
chapter 3 triangulations
§3.1 outerplanar triangulations
§3.2 planar triangulations
§3.3 triangulations on the disc

　　Combinatorics as a branch of mathematics studies the arts of counting. Enumeration occupies the foundation of combinatorics with a large range of applications not only in mathematics itself but also in many other disciplines. It is too broad a task to write a book to show the deep development in every corner from this aspect. This monograph is intended to provide a unified theory for those related to the enumeration of maps. .
　　For enumerating maps the first thing we have to know is the symmetry of a map. Or in other words, we have to know its automorphism group. In general, this is an interesting, complicated, and difficult problem. In order to do this, the first problem we meet is how to make a map considered without symmetry. Since the beginning of sixties when Tutte found a way of rooting on a map, the problem has been solved. This forms the basis of the enumerative theory of maps. As soon as the problem without considering the symmetry is solved for one kind of map, the general problem with symmetry can always, in principle, be solved from what we have known about the automorphism of a polyhedron, a synonym for a map, which can be determined efficiently according to another monograph of the present author.
　　Now, the problems facing us are how to find a functional equation satisfied by the enumerating function of one kind of map given and how to find a way to determine the coefficients in the power series form of the enumerating function by solving the equation. Then, a further problem is to investigate the stochastic behaviors of the kind of maps we have already enumerated.
　　For extracting an equation, a crucial trick is suitably to decompose the set of maps we are concerned with into several parts such that each of them can be generated by some operations on the set itself. The starting operations usually employed are the so called deletion and contraction of a specific edge properly chosen. Along this line one can see how different kinds of operations are constructed for enumerating a variety of types of maps. The way of decomposition is very closely related to the choice of parameters which the enumeration is according to. There are tricks which have to be exploited to avoid the complicatedness involved in deriving the functional equation from the decomposition.
　　As soon as a functional equation has been built up, the problem that follows is to find a suitable way to solve it, or to transform and simplify it for clarifying the solution. Here, we present a number of methods for solving the equations directly, or converting them into some special ones which are solvable in certain cases. The most interesting part is to try to find a way by which the Lagrangian inversion can be suitably applied for determining the coefficients in the power series form of the solution. Hopefully, many simpler formulae for enumerating a variety of maps have been obtained via a series of subtle treatments in this way. ..
　　In spite of whether the functional equation is completely solved or not, one is always allowed to investigate stochastic behaviors by estimating asymptotic properties of the solution as an enumerating function of certain kind of maps, when the order of maps is big enough, up to tending to infinity.
　　According to the basic theoretical idea as described above, the whole book is divided into three main parts. The first part, from Chapter 2 through Chapter 8, is on the ordinary theory of enumerating maps. The second, from Chapter 9 through Chapter 11, is on chromatic and dichromatic sums which can be seen as a kind of generalization of enumeration with much complication and much difficulty. And the third which consists of only one chapter, Chapter 12 is on the stochastic behaviors. Of course, Chapter I provides the necessary knowledge and basic techniques for the requirements of the whole book. In order to save space, the last section of each chapter is designed to be notes in which some historical remarks, new progress and unsolved problems with clues for possibly solving them are indicated in corresponding areas.
　　On this occasion, I should express my heartiest thanks to all those having made contributions themselves directly or indirectly to this book.
　　The initiation of this theory was established by Professor W. T. Tutte whose articles and directions were invoked for me to enter the field when I was working in the Department of Combinatorics and Optimization at the University of Waterloo, Canada, in the period of 1982-1984. Without these, I would be doing something else at the present time.
　　Many other friends of mine including Professors R. Cori, P.L. Hammer, D.M. Jackson, R. C. Mullin, R.C. Read, L.B. Richmond, P. Rosenstiehl, B. Simeone, T.T.S. Walsh, W. Xu and J. Yan are constantly concerned with me in material and spirit. For the final version, many people including J.L. Cai (PhD), Y.X. Chang (PhD), F.M. Dong (PhD), J.Q. Dong, R.X. Hao, Y.Q. Huang (PhD), S. Lawrencenko (PhD), A.P. Li (PhD), D.M. Li (PhD), X. Liu (PhD), Yi. Liu (PhD), T.Y. Liu, T.J. Lu (PhD), K. Ouyang (PhD), X.R. Sun (PhD), H. Ren, E.L. Wei, F.E. Wu (PhD) and M.L. Zheng (PhD) provide errata in part or whole.
　　The source Latex files for the whole book were typed and run on computers by my daughter Liu Ying.
　　Institutions including the Department of C & O, University of Waterloo, Canada; DIMACS and RUTCOR, Rutgers University, USA; the Department of Computer Science, the Department of Mathematics and the Department of Statistics, University of Rome 'La Sapienza', Italy; the Center of Applied Mathematics, E.H.E.S.S., France; the Department of Mathematics and Computer Science, University of Bordeaux I, France; and the Department of ECECS, University of Cincinnati, USA provided me the opportunities to visit with the hospitality. In particular, the Northern Jiaotong University where I am working now offers me favorable circumstances.
　　Last but not least, the partial financial support from the NSF in USA, the CNR in Italy and the NNSF in China should be especially acknowledged as well. ...
　　Y.P. Liu
　　Beijing, P.R. China.
　　May 1998