Introduction to Combinatorics
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BeschreibungThis concise text covers the fundamentals of pure mathematical combinatorics, including combinations, permutations, and more sophisticated counting techniques; generating functions; difference equations; generalizations; and special functions. It also presents graph-theoretic terminology and elementary graph theory as it arises in combinatorics. The authors provide brief introductions to more advanced ideas, such as Latin squares, block designs, matrices, and Polyaa (TM)s theory of counting. They also include exercises at both the elementary and advanced levels as well as additional exercises and solutions on a supplementary website.
InhaltsverzeichnisIntroduction 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 Polya-Redfield Method Introduction Basics of Groups Permutations and Colorings An Important Counting Theorem Polya 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.
PortraitW.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.
Pressestimmen... 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
Untertitel: 'Discrete Mathematics and Its A'. Sprache: Englisch.
Verlag: CRC PR INC
Erscheinungsdatum: September 2010
Seitenanzahl: 380 Seiten