Some of the problems are specially drawn from sources, such as the Amer ican Mathematical Monthly, Crux Mathematicorum, the College Mathe matics Journal and the Mathematical Magazine. Olympiad, the Asian Pacific Mathematics Olympiad, the USA Mathemat ical Olympiad and other Mathematical Olympiads from various nations. All in all, we have about 490 problems, which include combinatorial problems taken from mathematical competitions such as the IMO, the William Lowell Putnam Mathematical Competition, the Amer ican Invitational Mathematics Examination, the Singapore Mathematical v We have also included a wide range o f examples and exercises with various degrees of difficulty at the end of each chapter. By providing a plethora of carefully chosen examples, we hope that the applications of these principles as well as the techniques of generating functions and recurrence relations would be much more appreciated by the reader. In this book, we shall lay special emphasis on the importance of these principles, together with others, such as the Principle of Inclusion and Exclusion and the Pigeonhole Principle. We also find that students often neglect some o f the basic principles in combinatorics, such as the Addition Principle, the Multiplica tion Principle, the Principle of Complementation, the Injection Principle, and the Bijection Principle, perhaps due to their rather unsophisticated appearances. To achieve these objectives, we have tried to present the material very ex plicitly so that students with some mathematical maturity will find it very easy to read. Hence, in writing this book, we have two main objectives in mind: (1) it could be used as a text-book for undergraduate courses, and ( 2) it could be used for the training o f our International Mathematical Olympiad Teams. All along, we have been longing for a book that is suitable for our purposes. Both the authors have been involved in the teaching o f the subject as well as in the training of the Sin gapore International Mathematical Olympiad Teams for many years. Problems in Combinatorics are not only challenging for researchers, but also appear very frequently in various mathematical competitions, particularly the In ternational Mathematical Olympiad (IMO). The basic principles and tech niques taught in the course have found more and more applications in other fields, especially in computer science and operational research. Preface Over the years, Combinatorial Analysis has always been a popular course among our undergraduate students.
Printed in Singapore by Continental Press Pte Ltd This book, o r parts thereof, may not b e reproduced in any form o r by any means, electron ic o r m echanical, including photocopying, recording o r any inform ation stora ge and retrieval system now known o r to b e invented, without written perm ission from the Publisher. P R IN C IP L E S A N D T E C H N IQ U E S IN C O M B IN A T O R IC S Copyright © 1992 by W orld Scientific Publishing Co. P O B ox 128, Fairer Road, Singapore 9128 USA o ffice: Suite IB, 1060 Main Street, River Edge, NJ 07661 U K o ffice: 73 Lynton Mead, Totteridge, London N 20 8DH Published by W orld Scientific Publishing Co.
PRINCIPLES AND TECHNIQUES IN COMBINATORICSĬHEN CHUAN-CHONG and KOH KHEE-MENG Departm ent o f Mathematics National University o f Singaporeġ* World Scientific lb Singapore
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