Roberts F., Tesman B. Applied Combinatorics 3ed 2024

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Roberts F., Tesman B. Applied Combinatorics 3ed 2024

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Textbook in PDF format The original goal of writing this book was to introduce the reader to the tools of combinatorics from an applied point of view. This third edition of Applied Combinatorics was substantially rewritten. There are many new examples and exercises. References throughout the book to modern literature and real applications, a key feature of the book, have been updated and expanded. The exposition continues to be updated with each new edition, as the first edition was published 40 years ago. The emphasis on applications from computer science, genetics, experimental design, chemistry, scheduling, voting, and other topics remains a central feature of the book. Unique to the literature is that entire sections focus on applications such as switching functions, the use of enzymes to uncover unknown RNA chains, searching and sorting problems of information retrieval, construction of error-correcting codes, counting of chemical compounds, calculation of power in voting situations, and uses of Fibonacci numbers. There are entire sections on applications of recurrences involving convolutions, applications of eulerian chains, and applications of generating functions. The book continues to be based on the authors’ philosophy that the best way to learn mathematics is through problem-solving. Combinatorics can be a wonderful mechanism for introducing students to proofs. However, the book is not designed for an introduction to proofs course. The authors treat proofs as rather informal, and many of the harder proofs in the book are optional. Applied Combinatorics, Third Edition is divided into four parts. The first part introduces the basic tools of combinatorics and their applications. The remaining three parts are organized around the three basic problems of combinatorics: the counting problem, the existence problem, and the optimization problem. Most of the book is written for a first course on the topic at the undergraduate level. On the other hand, at a fast pace, there is more than enough material for a challenging graduate course. This book first appeared when courses on combinatorics were rare. We are pleased to think that, through its use, the book has helped to establish a key course in many colleges and universities throughout the world. We hope that this new edition will remain a valuable tool for instructors and students alike. Preface. Notation. What is Combinatorics? The Three Problems of Combinatorics. The History and Applications of Combinatorics. References. The Basic Tools of Combinatorics Basic Counting Rules The Product Rule. The Sum Rule. Permutations. Complexity of Computation. r-Permutations. Subsets. r-Combinations. Probability. Sampling with Replacement. Occupancy Problems. Multinomial Coefficients. Complete Digest by Enzymes. Permutations with Classes of Indistinguishable Objects Revisited. The Binomial Expansion. Power in Simple Games. Generating Permutations and Combinations. Inversion Distance between Permutations and the Study of Mutations. Good Algorithms. Pigeonhole Principle and Its Generalizations. Additional Exercises. References. Introduction to Graph Theory Fundamental Concepts. Connectedness. Graph Coloring and Its Applications. Chromatic Polynomials. Trees. Applications of Rooted Trees to Searching, Sorting, and Phylogeny Reconstruction. References. Relations Relations. Order Relations and Their Variants. Linear Extensions of Partial Disorders. Lattices and Boolean Algebras. References. The Counting Problem Generating Functions and Their Applications Examples of Generating Functions. Operating on Generating Functions. Applications to Counting. The Binomial Theorem. Exponential Generating Functions and Generating Functions for Permutations. Probability Generating Functions. The Coleman and Banzhaf Power Indices. References. Recurrence Relations Some Examples. The Method of Characteristic Roots. Solving Recurrences Using Generating Functions. Some Recurrences Involving Convolutions. References. The Principle of Inclusion and Exclusion The Principle and Some of Its Applications. The Number of Objects Having Exactly m Properties. References. The Pólya Theory of Counting Equivalence Relations. Permutation Groups. Burnside’s Lemma. Distinct Colorings. The Cycle Index. Pólya’s Theorem. References. The Existence Problem Combinatorial Designs Block Designs. Latin Squares. Finite Fields and Complete Orthogonal Families of Latin Squares. Balanced Incomplete Block Designs. Finite Projective Planes. References. Coding Theory Information Transmission. Encoding and Decoding. Error-Correcting Codes. Linear Codes. The Use of Block Designs to Find Error-Correcting Codes. References. Existence Problems in Graph Theory Depth-First Search: A Test for Connectedness. The One-Way Street Problem. Eulerian Chains and Paths. Applications of Eulerian Chains and Paths. Hamiltonian Chains and Paths. Applications of Hamiltonian Chains and Paths. References. Combinatorial Optimization Matching and Covering. Some Matching Problems. Some Existence Results: Bipartite Matching and Systems of Distinct Representatives. The Existence of Perfect Matchings for Arbitrary Graphs. Maximum Matchings and Minimum Coverings. Finding a Maximum Matching. Matching as Many Elements of X as Possible. Maximum-Weight Matching. Stable Matchings. References. Optimization Problems for Graphs and Networks Minimum Spanning Trees. The Shortest Route Problem. Network Flows. Minimum-Cost Flow Problems. References. Author Index. Subject Index