Preface to the First Edition

Abstract

Combinatorial analysis or combinatorics, for short, deals with enumerative problems where one must answer the question “How many ?” or “In how many ways?” Other problems are concerned with the existence of certain combinatorial objects subject to various constraints. These kinds of problems are considered in this book. Combinatorial problems, methods and graphical models are abundant in many areas ranging from engineering and financial science to humanitarian disciplines like sociology, psychology,medicine and social sciences, not tomentionmathematics and computer science. As parts of discrete mathematics, combinatorics and graph theory have become indispensable parts of introductory and advanced mathematical training for everyone dealing not only with quantitative but also with qualitative data. Moreover, combinatorics and graph theory have a remarkable and uncommon feature—tobegin its study, oneneedsnobackgroundbut elementary algebra and common sense. Even simple combinatorial problems often lead to interesting, sometimes difficult questions and allow an instructor to introduce various important mathematical ideas and concepts and to show the nature of mathematical reasoning and proof. These qualities make combinatorics and graph theory an excellent choice for an introductory mathematical class for students of any age, level and major. This is a text for a one-semester course in combinatorics with elements of graph theory. It can be used in twomodes. The first three chapters cover an introductorymaterial and can be (and have actually been) used for an undergraduate class in combinatorics and/or discretemathematics, as well as for a problem-solving seminar aimed at undergraduate and even motivated high-school students. Chapters 4 and 5 are of more advanced level and the whole book includes enough material for an entry-level graduate course in combinatorics. For the mathematically inclined reader, the material has been developed systematically and includes all the proofs. After this book, the reader can study more advanced courses, e. g. [1, 9, 10, 22, 51]. At the same time, the reader who is primarily interested in applying combinatorial methods can skip (most of) the proofs and concentrate on problems and methods of their solution. In Chapter 1 we introduce basic combinatorial concepts, such as the sum and product rules, combinations, permutations, and arrangements with and without repetition. Various particular elementarymethods of solving combinatorial problems are also considered throughout the book, such as, for instance, the trajectory method in Section 1.4 or Ferrers diagrams in Section 4.4. In Section 1.6 we apply the methods of Sections 1.1–1.5 to develop the elementary probability theory for random experiments with finite sample spaces. Our goal in this section is not to give a systematic exposition of probability theory, but rather to show some meaningful applications of the combinatorial methods developed earlier. Chapter 2 contains an introduction to graph theory. After setting up the basic vocabulary in Sections 2.1–2.2, in the next three sections we study properties of trees,