Introduction to Combinatorics: (Discrete Mathematics and Its Applications)

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 Pólya 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.

Table of Contents:
Introduction Some 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 Theorem The Principle of Inclusion and Exclusion Unions of Events The Principle Combinations with Limited Repetitions Derangements Generating Functions and Recurrence Relations Generating Functions Recurrence Relations From Generating Function to Recurrence Exponential Generating Functions Catalan, Bell and Stirling Numbers Introduction Catalan Numbers Stirling Numbers of the Second Kind Bell Numbers Stirling Numbers of the First Kind Computer Algebra and Other Electronic Systems Symmetries and the Pólya–Redfield Method Introduction Basics of Groups Permutations and Colorings An Important Counting Theorem Pólya and Redfield’s Theorem Introduction to Graph Theory Degrees Paths and Cycles in Graphs Maps and Graph Coloring Further Graph Theory Euler Walks and Circuits Application 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 Designs New Designs from Old Difference Method Linear Algebra Methods in Combinatorics Recurrences 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 Exercises Hints for Problems Solutions to Problems References Index Exercises and Problems appear at the end of each chapter.

About the Author :
W.D. Wallis is Emeritus Professor of Mathematics at Southern Illinois University. His research interests include combinatorial designs, Latin squares, graph labeling, one-factorizations, and intelligent networks. Dr. Wallis is the author of Introduction to Combinatorial Designs, Second Edition (CRC Press, 2007). J.C. George is an assistant professor of mathematics in the Division of Mathematics and Natural Sciences at Gordon College in Barnesville, Georgia. His research interests include one-factorizations, graph products, and the relationships of algebraic structures to combinatorial objects.

Review :
… thoughtfully written, contain[s] plenty of material and exercises … provides numerous fragments of Mathematica code and this is a nice touch. … —MAA Reviews, February 2011