Combinatorial Methods with Computer Applications

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.