Combinatorial Methods with Computer Applications

Combinatorial Methods with Computer Applications

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Combinatorial Methods with Computer Applications provides in-depth coverage of recurrences, generating functions, partitions, and permutations, along with some of the most interesting graph and network topics, design constructions, and finite geometries. Requiring only a foundation in discrete mathematics, it can serve as the textbook in a combinatorial methods course or in a combined graph theory and combinatorics course. After an introduction to combinatorics, the book explores six systematic approaches within a comprehensive framework: sequences, solving recurrences, evaluating summation expressions, binomial coefficients, partitions and permutations, and integer methods. The author then focuses on graph theory, covering topics such as trees, isomorphism, automorphism, planarity, coloring, and network flows. The final chapters discuss automorphism groups in algebraic counting methods and describe combinatorial designs, including Latin squares, block designs, projective planes, and affine planes. In addition, the appendix supplies background material on relations, functions, algebraic systems, finite fields, and vector spaces. Paving the way for students to understand and perform combinatorial calculations, this accessible text presents the discrete methods necessary for applications to algorithmic analysis, performance evaluation, and statistics as well as for the solution of combinatorial problems in engineering and the social sciences.show more

Product details

  • Hardback | 664 pages
  • 180.34 x 254 x 38.1mm | 1,315.41g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 287 black & white illustrations
  • 1584887435
  • 9781584887430
  • 1,417,279

Review quote

... The book is very carefully written and might be a good starting point for undergraduate students. ... -Zentralblatt MATH 1168 I recently got [this] book on combinatorics and applications to computer science, and I like it so much that I am trying to re-shape some of the discrete maths courses I teach so that I could use it. I liked particularly [the] section on asymptotics, which is much more accessible for my undergrads than Graham, Knuth, and Patashnik. -Josef Lauri, University of Maltashow more

Table of contents

PREFACE INTRODUCTION TO COMBINATORICS Objectives of Combinatorics Ordering and Selection Some Rules of Counting Counting Selections Permutations Graphs Number-Theoretic Operations Combinatorial Designs SEQUENCES Sequences as Lists Recurrences Pascal's Recurrence Differences and Partial Sums Falling Powers Stirling Numbers: A Preview Ordinary Generating Functions Synthesizing Generating Functions Asymptotic Estimates SOLVING RECURRENCES Types of Recurrences Finding Generating Functions Partial Fractions Characteristic Roots Simultaneous Recursions Fibonacci Number Identities Nonconstant Coefficients Divide-and-Conquer Relations EVALUATING SUMS Normalizing Summations Perturbation Summing with Generating Functions Finite Calculus Iteration and Partitioning of Sums Inclusion-Exclusion SUBSETS AND BINOMIALS Binomial Coefficient Identities Binomial Inversion Operation Applications to Statistics The Catalan Recurrence PARTITIONS AND PERMUTATIONS Stirling Subset Numbers Stirling Cycle Numbers Inversions and Ascents Derangements Exponential Generating Functions Posets and Lattices INTEGER OPERATIONS Euclidean Algorithm Chinese Remainder Theorem Polynomial Divisibility Prime and Composite Moduli Euler Phi-Function The Mobius Function GRAPH FUNDAMENTALS Regular Graphs Walks and Distance Trees and Acyclic Digraphs Graph Isomorphism Graph Automorphism Subgraphs Spanning Trees Edge Weights Graph Operations GRAPH THEORY TOPICS Traversability Planarity Coloring Analytic Graph Theory Digraph Models Network Flows Topological Graph Theory GRAPH ENUMERATION Burnside-Polya Counting Burnside's Lemma Counting Small Simple Graphs Partitions of Integers Calculating a Cycle Index General Graphs and Digraphs DESIGNS Latin Squares Block Designs Classical Finite Geometries Projective Planes Affine Planes APPENDIX Relations and Functions Algebraic Systems Finite Fields and Vector Spaces BIBLIOGRAPHY General Reading References SOLUTIONS AND HINTS INDICES A Glossary appears at the end of each chapter.show more

About Jonathan L. Gross

Columbia University, New York, USAshow more

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