Introduction to CombinatoricsThis 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. |
Contents
Existence | 23 |
Sequences and Partial Orders | 40 |
Ramsey Theory | 49 |
Enumeration | 73 |
Recurrence Relations and Explicit Formulas | 82 |
Permutations and Tableaux | 111 |
The Pólya Theory of Counting | 117 |
Construction | 135 |
Designs | 154 |
Big Designs | 178 |
| 187 | |
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Common terms and phrases
2-coloring A₁ algebraic algorithm antichain BIBD bijection blocks Burnside's lemma called Chapter codeword columns combinatorial complete graph contact number contains cycle index define determine Dilworth's lemma edges equation equivalence classes example Exercise exists Ferrers diagram Figure follows formula function f graph G graph theory Hadamard matrix Hamming code homomorphism identity increasing subsequence inequality infinitary infinite inverse isomorphic k-subsets labeled Latin squares lattice points length linear maximum number modulo MOLS monochromatic multiplication nonisomorphic graphs number of elements number of functions obtain one-to-one Open Problem ordered pairs orthogonal partial order permutation pigeonhole principle plane of order projective plane Proof prove Putnam Competition quadratic Ramsey numbers Ramsey theory Ramsey's theorem reader real numbers recurrence relation Section sequence Show Steiner system Stirling numbers subgroup subsets Suppose symmetric total number unlabeled upper bound values vector vertex weight Young tableaux