THEORY AND PROBLEMS OF

COMBINATORICS

including concepts of GRAPH THEORY

V.K.Balakrishnan

Contents

Chapter 1            BASIC TOOLS

1.1   The Sum Rule and the Product Rule
1.2   Permutations and Combinations
1.3   The Pigeonhole Principle

Solved Problems
      The Sum and Product Rules
      Permutations and Combinations
      The Pigeonhole Principle
      Ramsey Numbers
      Catalan Numbers
      Stirling Numbers

Chapter 2 FURTHER BASIC TOOLS 2.1 Generalized Permutations and Combinations 2.2 Sequences and Selections 2.3 The Inclusion-Exclusion Principle 2.4 Systems of Distinct Representatives (SDR) Solved Problems Generalized Permutations and Combinations Sequences and Selections The inclusion - exclusion Principle Derangements and Other Constrained Arrangements Combinatorial Number Theory Generalized Inclusion-Exclusion Principle The Permanent of a Matrix Rook Polynomials and Hit Polynomials Systems of Distinct Representatives (SDR) and Coverings Spernerís Theorem and Symmetric Chain Decomposition Partially Ordered Sets and Dilworth's Theorem
Chapter 3 GENERATING FUNCTIONS AND RECURRENCE RELATIONS 3.1 Ordinary and Exponential Generating Functions 3.2 Partitions of a Positive Integer 3.3 Recurrence Relations 3.4 Algebraic Solution of Linear Recurrence Relations with Constant Coefficients 3.5 Solution of Recurrence Relations Using Generating Functions Solved Problems Ordinary Generating Functions Partitions of Integers and Their Generating Functions Exponential Generating Functions Recurrence Relations and Associated Generating Functions H-Tableaux and Young Tableaux
Chapter 4 GROUP THEORY IN COMBINATORICS 4.1 The Burnside-Frobenius Theorem 4.2 Permutation Groups and Their Cycle Indices 4.3 Polya's Enumeration Theorems Solved Problems The Burnside-Frobenius Theorem Permutation Groups and Their Cycle Indices Polya's Enumeration Theorems
Appendix GRAPH THEORY
GLOSSARY
LIST OF SYMBOLS
FURTHER READING
INDEX

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