Introduction to Combinatorics

1996 | 201 Pages | ISBN: 0471154083 | PDF | 4 MB

This gradual, systematic introduction to the main concepts of combinatorics is the ideal text for advanced undergraduate and early graduate courses in this subject. Each of the book's three sections--Existence, Enumeration, and Construction--begins with a simply stated first principle, which is then developed step by step until it leads to one of the three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Polya's graph enumeration formula, and Leech's 24-dimensional lattice. Along the way, Professor Martin J. Erickson introduces fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise new and innovative questions and observations. His carefully chosen end-of-chapter exercises demonstrate the applicability of combinatorial methods to a wide variety of problems, including many drawn from the William Lowell Putnam Mathematical Competition. Many important combinatorial methods are revisited several times in the course of the text--in exercises and examples as well as theorems and proofs. This repetition enables students to build confidence and reinforce their understanding of complex material. Mathematicians, statisticians, and computer scientists profit greatly from a solid foundation in combinatorics. Introduction to Combinatorics builds that foundation in an orderly, methodical, and highly accessible manner. Content: Chapter 1 Preliminaries: Set Theory, Algebra, and Number Theory (pages 1-22): Chapter 2 The Pigeonhole Principle (pages 25-39): Chapter 3 Sequences and Partial Orders (pages 40-48): Chapter 4 Ramsey Theory (pages 49-71): Chapter 5 The Fundamental Counting Problem (pages 75-81): Chapter 6 Recurrence Relations and Explicit Formulas (pages 82-110): Chapter 7 Permutations and Tableaux (pages 111-116): Chapter 8 The Polya Theory of Counting (pages 117-133): Chapter 9 Codes (pages 137-153): Chapter 10 Designs (pages 154-177): Chapter 11 Big Designs (pages 178-186):

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