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Full Description
Accessible to undergraduate students, Introduction to Combinatorics presents approaches for solving counting and structural questions. It looks at how many ways a selection or arrangement can be chosen with a specific set of properties and determines if a selection or arrangement of objects exists that has a particular set of properties.To give students a better idea of what the subject covers, the authors first discuss several examples of typical combinatorial problems. They also provide basic information on sets, proof techniques, enumeration, and graph theory-topics that appear frequently throughout the book. The next few chapters explore enumerative ideas, including the pigeonhole principle and inclusion/exclusion. The text then covers enumerative functions and the relations between them. It describes generating functions and recurrences, important families of functions, and the theorems of Polya and Redfield. The authors also present introductions to computer algebra and group theory, before considering structures of particular interest in combinatorics: graphs, codes, Latin squares, and experimental designs. The last chapter further illustrates the interaction between linear algebra and combinatorics. Exercises and problems of varying levels of difficulty are included at the end of each chapter.Ideal for undergraduate students in mathematics taking an introductory course in combinatorics, this text explores the different ways of arranging objects and selecting objects from a set. It clearly explains how to solve the various problems that arise in this branch of mathematics.
Contents
IntroductionSome Combinatorial Examples Sets, Relations and Proof Techniques Two Principles of Enumeration Graphs Systems of Distinct Representatives Fundamentals of Enumeration Permutations and Combinations Applications of P(n, k) and (n k)Permutations and Combinations of Multisets Applications and Subtle Errors Algorithms The Pigeonhole Principle and Ramsey's Theorem The Pigeonhole Principle Applications of the Pigeonhole Principle Ramsey's Theorem - the Graphical Case Ramsey Multiplicity Sum-Free Sets Bounds on Ramsey Numbers The General Form of Ramsey's TheoremThe Principle of Inclusion and Exclusion Unions of Events The Principle Combinations with Limited Repetitions DerangementsGenerating Functions and Recurrence Relations Generating Functions Recurrence Relations From Generating Function to Recurrence Exponential Generating FunctionsCatalan, Bell and Stirling Numbers Introduction Catalan Numbers Stirling Numbers of the Second Kind Bell Numbers Stirling Numbers of the First KindComputer Algebra and Other Electronic SystemsSymmetries and the Polya-Redfield Method Introduction Basics of Groups Permutations and Colorings An Important Counting TheoremPolya and Redfield's TheoremIntroduction to Graph Theory Degrees Paths and Cycles in Graphs Maps and Graph ColoringFurther Graph TheoryEuler Walks and CircuitsApplication of Euler Circuits to Mazes Hamilton Cycles Trees Spanning Trees Coding Theory Errors; Noise The Venn Diagram Code Binary Codes; Weight; Distance Linear Codes Hamming Codes Codes and the Hat Problem Variable-Length Codes and Data Compression Latin Squares Introduction Orthogonality Idempotent Latin Squares Partial Latin Squares and Subsquares Applications Balanced Incomplete Block Designs Design Parameters Fisher's Inequality Symmetric Balanced Incomplete Block DesignsNew Designs from Old Difference MethodLinear Algebra Methods in CombinatoricsRecurrences Revisited State Graphs and the Transfer Matrix Method Kasteleyn's Permanent Method Appendix 1: Sets; Proof Techniques Appendix 2: Matrices and Vectors Appendix 3: Some Combinatorial People Solutions to Set A ExercisesHints for ProblemsSolutions to ProblemsReferences IndexExercises and Problems appear at the end of each chapter.
