TAILIEUCHUNG - generatingfunctionology

.Preface This book is about generating functions and some of their uses in discrete mathematics. The subject is so vast that I have not attempted to give a comprehensive discussion. Instead I have tried only to communicate some of the main ideas. Generating functions are a bridge between discrete mathematics, on the one hand, and continuous analysis (particularly complex variable theory) on the other. It is possible to study them solely as tools for solving discrete problems. | generatingfunctionology Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia Pennsylvania Copyright 1990 and 1994 by Academic Press Inc. All rights reserved. This Internet Edition may be reproduced for any valid educational purpose of an institution of higher learning in which case only the reasonable costs of reproduction may be charged. Reproduction for profit or for any commercial purposes is strictly prohibited. vi Preface This book is about generating functions and some of their uses in discrete mathematics. The subject is so vast that I have not attempted to give a comprehensive discussion. Instead I have tried only to communicate some of the main ideas. Generating functions are a bridge between discrete mathematics on the one hand and continuous analysis particularly complex variable theory on the other. It is possible to study them solely as tools for solving discrete problems. As such there is much that is powerful and magical in the way generating functions give unified methods for handling such problems. The reader who wished to omit the analytical parts of the subject would skip chapter 5 and portions of the earlier material. To omit those parts of the subject however is like listening to a stereo broadcast of say Beethoven s Ninth Symphony using only the left audio channel. The full beauty of the subject of generating functions emerges only from tuning in on both channels the discrete and the continuous. See how they make the solution of difference equations into child s play. Then see how the theory of functions of a complex variable gives virtually by inspection the approximate size of the solution. The interplay between the two channels is vitally important for the appreciation of the music. In recent years there has been a vigorous trend in the direction of finding bijective proofs of combinatorial theorems. That is if we want to prove that two sets have the same cardinality then we should be able to do it by exhibiting

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