Combinatorial Methods with Computer Applications

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.

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.

PREFACEINTRODUCTION TO COMBINATORICSObjectives of CombinatoricsOrdering and SelectionSome Rules of CountingCounting SelectionsPermutationsGraphsNumber-Theoretic OperationsCombinatorial DesignsSEQUENCESSequences as ListsRecurrencesPascal's RecurrenceDifferences and Partial SumsFalling PowersStirling Numbers: A PreviewOrdinary Generating FunctionsSynthesizing Generating FunctionsAsymptotic EstimatesSOLVING RECURRENCESTypes of RecurrencesFinding Generating FunctionsPartial FractionsCharacteristic RootsSimultaneous RecursionsFibonacci Number IdentitiesNonconstant CoefficientsDivide-and-Conquer RelationsEVALUATING SUMSNormalizing SummationsPerturbationSumming with Generating FunctionsFinite CalculusIteration and Partitioning of SumsInclusion-ExclusionSUBSETS AND BINOMIALSBinomial Coefficient IdentitiesBinomial Inversion OperationApplications to StatisticsThe Catalan RecurrencePARTITIONS AND PERMUTATIONSStirling Subset NumbersStirling Cycle NumbersInversions and AscentsDerangementsExponential Generating FunctionsPosets and LatticesINTEGER OPERATIONSEuclidean AlgorithmChinese Remainder TheoremPolynomial DivisibilityPrime and Composite ModuliEuler Phi-FunctionThe Möbius FunctionGRAPH FUNDAMENTALSRegular GraphsWalks and DistanceTrees and Acyclic DigraphsGraph IsomorphismGraph AutomorphismSubgraphsSpanning TreesEdge WeightsGraph OperationsGRAPH THEORY TOPICSTraversabilityPlanarityColoringAnalytic Graph TheoryDigraph ModelsNetwork FlowsTopological Graph TheoryGRAPH ENUMERATIONBurnside-Pólya CountingBurnside's LemmaCounting Small Simple GraphsPartitions of IntegersCalculating a Cycle IndexGeneral Graphs and DigraphsDESIGNSLatin SquaresBlock DesignsClassical Finite GeometriesProjective PlanesAffine PlanesAPPENDIXRelations and FunctionsAlgebraic SystemsFinite Fields and Vector SpacesBIBLIOGRAPHYGeneral ReadingReferencesSOLUTIONS AND HINTSINDICESA Glossary appears at the end of each chapter.