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This book provides an introduction to the main concepts of combinatorics, features fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations.is the ideal text for advanced undergraduate and early graduate courses in this subject. The Second Edition contains over fifty new examples that illustrate important combinatorial concepts and range from the routine (i.e. special kinds of sets, functions, and sequences) to the advanced (i.e. the SET game, the Gitterpunktproblem, and enumeration of partial orders). The tables and references are been updated throughout, reflecting advances in Ramsey numbers and Thomas Hales' solution of Kepler's conjecture). In addition, many exciting new computer programs and exercises have been incorporated to help readers understand and apply combinatorial techniques and ideas. The author has now made it possible for readers to encode and execute programs for formulas that were previously inaccessible, allowing for a deeper, investigative study of combinatorics. Each of the book's three sections, Existence, Enumeration, and Construction, begin 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. Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, enabling readers to build confidence and reinforce their understanding of complex material.
MARTIN J. ERICKSON, PhD, is Professor in the Department of Mathematics at Truman State University. The author of numerous books, including Mathematics for the Liberal Arts (Wiley), he is a member of the American Mathematical Society, Mathematical Association of America, and American Association of University Professors.
Table of Contents
1 Basic Counting Methods 1
1.1 The multiplication principle 1
1.2 Permutations 4
1.3 Combinations 6
1.4 Binomial coefficient identities 10
1.5 Distributions 19
1.6 The principle of inclusion and exclusion 23
1.7 Fibonacci numbers 31
1.8 Linear recurrence relations 33
1.9 Special recurrence relations 41
1.10 Counting and number theory 45
2 Generating Functions 53
2.1 Rational generating functions 53
2.2 Special generating functions 63
2.3 Partition numbers 76
2.4 Labeled and unlabeled sets 80
2.5 Counting with symmetry 86
2.6 Cycle indexes 93
2.7 Pólya’s theorem 96
2.8 The number of graphs 98
2.9 Symmetries in domain and range 102
2.10 Asymmetric graphs 103
3 The Pigeonhole Principle 107
3.1 Simple examples 107
3.2 Lattice points, the Gitterpunktproblem, and SET® 110