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内容简介
This third edition of Introductory Combinatorics contains extensive rewriting of some sections and the inclusion of some new material and exercises.There is enough material in this third edition for a two semester course. A first semester could have an emphasis on counting and a second semester an emphasis on graph theory. It is difficult to assess the prerequisites for this book. Perhaps they can be best described as the mathematical maturity achieved by the successful completion of the calculus sequence and an elementary course on linear algebra. Use of calculus is minimal, and the references to linear algebra axe few and should not cause any problem to those not familiar with it.
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目录
目 录 Chapter 1. What is Combinatorics 1.1 Example. Perfect covers of chessboards 1.2 Example. Cutting a cube 1.3 Example. Magic squares 1.4 Example. The 4-color problern 1.5 Example. The problem of the 36 officers 1.6 Example. Shortest-route problem 1.7 Example. The game of Nim 1.8 Exercises Chapter 2. The Pigeonhole Principle 2.1 Pigeonhole principle: Simple form 2.2 Pigeonhole principle: Strong form 2.3 A theorem of Ramsey 2.4 Exercises Chapter 3. Permutations and Combinations 3.1 Two basic counting principles 3.2 Permutations of sets 3.3 Combinations of sets 3.4 Permutations of multisets 3.5 Combinations of multisets 3.6 Exercises Chapter 4. Generating Permutations and Combinations 4.1 Generating permutations 4.2 Inversions in permutations 4.3 Generating combinations 4.5 Partial orders and equivalence relations 4.6 Exercises Chapter 5. The Binomial Coemcients 5.1 Pascal''''s formula 5.2 The binomial theorem 5.3 Identities 5.4 Unimodality of binomial coemcients 5.5 The multinomial theorem 5.6 Newton''''s binomial theorem 5.7 More on partially ordered sets 5.8 Exercises ChHpter 6. The Inclusion-Exclusion Principle and Applicutions 6.1 The inclusion-exclusion principle 6.2 Combinations with repetition 6.3 Derangements 6.4 Permutations with forbidden positions 6.5 Another forbidden position problem 6.6 Exercises Chapter 7. necurrence Helations and thenerating Functions 7.1 Some number sequences 7.2 Linear homogeneous recurrence relations 7.3 Non-homogeneous recurrence relations 7.4 Generating functions 7.5 Recurrences and generating functions 7.6 A geometry example 7.7 Exponential generating functions 7.8 Exercises Chnpter 8. Specinl Counting Sequences 8.1 Catalan numbers 8.2 Difference sequences and Stirling numbers 8.3 Partition numbers 8.4 A geometric problem 8.5 Exercises Chapter 9. Matchings in BipHrtite Oraphs 9.1 General problem formulation 9.2 Matchings 9.3 Systems of distinct representatives 9.4 Stable Inarriages 9.5 Exercises Chapter 10. Combinatorial nesigns 10.1 Modular arithmetic 10.2 Block designs 10.3 Steiner triple systems 10.4 Latin squares 10.5 Exercises Chapter 11. Introduction to Orsiph Theory 11.1 Basic properties 11.2 Eulerian trails 11.3 Hamilton chains and cycles 11.4 Bipartite multigraphs 11.5 Trees 11.6 The Shannon switching game 11.7 More on trees 11.8 Exercises Chapter 12. nigrHphs Hnd Networks 12.1 Digraphs 12.2 Networks 12.3 Exercises Chupter 13. More on Oruph Theory 13.1 Chromatic number 13.2 Plane and planar graphs 13.3 A 5-color theorem 13.4 Independence number and clique number 13.5 Connectivity 13.6 Exercises Chapter 14. Polya Counting 14.1 Permutation and symmetry groups 14.2 Burnside''''s theorem 14.3 Polya''''s counting formula 14.4 Exercises Answers sind Hints to Exercises Bibliography Index
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