Introduction to Combinatorics

What Is Combinatorics Anyway? Broadly speaking, combinatorics is the branch of mathematics dealing with different ways of selecting objects from a set or arranging objects. It tries to answer two major kinds of questions, namely, counting questions: how many ways can a selection or arrangement be chosen with a particular set of properties; and structural questions: does there exist a selection or arrangement of objects with a particular set of properties? The authors have presented a text for students at all levels of preparation. For some, this will be the first course where the students see several real proofs. Others will have a good background in linear algebra, will have completed the calculus stream, and will have started abstract algebra. The text starts by briefly discussing several examples of typical combinatorial problems to give the reader a better idea of what the subject covers. The next chapters explore enumerative ideas and also probability. It then moves on to enumerative functions and the relations between them, and generating functions and recurrences., Important families of functions, or numbers and then theorems are presented. Brief introductions to computer algebra and group theory come next. Structures of particular interest in combinatorics: posets, graphs, codes, Latin squares, and experimental designs follow. The authors conclude with further discussion of the interaction between linear algebra and combinatorics. Features Two new chapters on probability and posets. Numerous new illustrations, exercises, and problems. More examples on current technology use A thorough focus on accuracy Three appendices: sets, induction and proof techniques, vectors and matrices, and biographies with historical notes, Flexible use of MapleTM and MathematicaTM

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 Probability Introduction Some Definitions and Easy Examples Events and Probabilities Three Interesting Examples Probability Models Bernoulli Trials The Probabilities in Poker The Wild Card Poker Paradox 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 P´olya-Redfield Method Introduction Basics of Groups Permutations and Colorings An Important Counting Theorem P´olya and Redfield’s Theorem Partially-Ordered Sets Introduction Examples and Definitions Bounds and lattices Isomorphism and Cartesian products Extremal set theory: Sperner’s and Dilworth’s theorems 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 Methods Linear Algebra Methods in Combinatorics Recurrences Revisited State Graphs and the Transfer Matrix Method Kasteleyn’s Permanent Method Appendix 1: Sets; Proof Techniques7 Appendix 2: Matrices and Vectors Appendix 3: Some Combinatorial People