Preface
1 What Is Combinatorics？
1.1 Example:Perfect Covers of Chessboards
1.2 Example:Magic Squares
1.3 Example:The Four-Color Problem
1.4 Example:The Problem of the 36 OfFicers
1.5 ,Example:Shortest-Route Problem
1.6 Example:Mutually Overlapping Circles
1.7 Example:The Game of Nim
1.8 Exercises
2 Permutations and Combinations
2.1 Four Basic Counting Principles
2.2 Permutations of Sets
2.3 Combinations（Subsets）of Sets
2.4 Permutations of Multisets
2.5 Combinations of Multisets
2.6 Finite Probability
2.7 Exercises
3 The Pigeonhole Principle
3.1 Pigeonhole Principle:Simple Form
3.2 Pigeonhole Principle:Strong Form
3.3 A Theorem of Ramsey
3.4 Exercises
4 Generating Permutations and Combinations
4.1 Generating Permutations
4.2 Inversions in Permutations
4.3 Generating Combinations
4.4 Generating r-Subsets
4.5 Partial Orders and Equivalence Relations
4.6 Exercises
5 The Binomial Coefficients
5.1 Pascal’s Triangle
5.2 The Binomial Theorem
5.3 Unimodality of Binomial Coefficients
5.4 The Multinomial Theorem
5.5 Newton’s Binomial Theorem
5.6 More on Partially Ordered Sets
5.7 Exercises
6 The Inclusion-Exclusion Principle and Applications
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 M6bius Inversion
6.7 Exercises
7 Recurrence Relations and Generating Functions
7.1 Some Number Sequences
7.2 Generating Functions
7.3 Exponential Generating Functions
7.4 Solving Linear Homogeneous Recurrence Relations
7.5 Nonhomogeneous Recurrence Relations
7.6 A Geometry Example
7.7 Exercises
8 Special Counting Sequences
8.1 Catalan Numbers
8.2 Difference Sequences and Stirling Numbers
8.3 Partition Numbers
8.4 A Geometric Problem
8.5 Lattice Paths and Schr6der Numbers
8.6 Exercises
9 Systems of Distinct Representatives
9.1 General Problem Formulation
9.2 Existence of SDRs
9.3 Stable Marriages
9.4 Exercises
10 Combinatorial Designs
10.1 Modular Arithmetic
10.2 Block Designs
10.3 Steiner Triple Systems
10.4 Latin Squares
10.5 Exercises
11 Introduction to Graph Theory
11.1 Basic Properties
11.2 Eulerian Trails
11.3 Hamilton Paths and Cycles
11.4 Bipartite Multigraphs
11.5 Trees
11.6 The Shannon Switching Game
11.7 More on Trees
11.8 Exercises
12 More on Graph Theory
12.1 Chromatic Number
12.2 Plane and Planar Graphs
12.3 A Five-Color Theorem
12.4 lndependence Number and Clique Number
12.5 Matching Number
12.6 Connectivity
12.7 Exercises
13 Digraphs and Networks
13.1 Digraphs
13.2 Networks
13.3 Matchings in Bipartite Graphs Revisited
13.4 Exercises
14 Polya Counting
14.1 Permutation and Symmetry Groups
14.2 Burnside’s Theorem
14.3 Polya’s Counting Formula
14.4 Exercises
Answers and Hints to Exercises
Bibliography
Index
