Introduction to Combinatorics

Introduction to Combinatorics

1st Edition - April 28, 1972

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  • Authors: Gerald Berman, K. D. Fryer
  • eBook ISBN: 9781483273822

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Description

Introduction to Combinatorics focuses on the applications, processes, methodologies, and approaches involved in combinatorics or discrete mathematics. The book first offers information on introductory examples, permutations and combinations, and the inclusion-exclusion principle. Discussions focus on some applications of the inclusion-exclusion principle, derangements, calculus of sets, permutations, combinations, Stirling's formula, binomial theorem, regions of a plane, chromatic polynomials, and a random walk. The text then examines linear equations with unit coefficients, recurrence relations, and generating functions. Topics include derivatives and differential equations, solution of difference equations by means of generating functions, recurrence relations, summation method, difference methods, combinations with repetitions, solutions bounded below, and solutions bounded above and below. The publication takes a look at generating functions and difference equations, ramifications of the binomial theorem, finite structures, coloring problems, maps on a sphere, and geometry of the plane. The manuscript is a valuable reference for researchers interested in combinatorics.

Table of Contents


  • Preface

    Acknowledgments

    1. Introductory Examples

    1.1 A Simple Enumeration Problem

    1.2 Regions of a Plane

    1.3 Counting Labeled Trees

    1.4 Chromatic Polynomials

    1.5 Counting Hairs

    1.6 Evaluating Polynomials

    1.7 A Random Walk

    Part I. Enumeration

    2. Permutations and Combinations

    2.1 Permutations

    2.2 r-Arrangements

    2.3 Combinations

    2.4 The Binomial Theorem

    2.5 The Binomial Coefficients

    2.6 The Multinomial Theorem

    2.7 Stirling's Formula

    3. The Inclusion-Exclusion Principle

    3.1 A Calculus of Sets

    3.2 The Inclusion-Exclusion Principle

    3.3 Some Applications of the Inclusion-Exclusion Principle

    3.4 Derangements

    4. Linear Equations with Unit Coefficients

    4.1 Solutions Bounded Below

    4.2 Solutions Bounded Above and Below

    4.3 Combinations with Repetitions

    5. Recurrence Relations

    5.1 Recurrence Relations

    5.2 Solution by Iteration

    5.3 Difference Methods

    5.4 A Fibonacci Sequence

    5.5 A Summation Method

    5.6 Chromatic Polynomials

    6. Generating Functions

    6.1 Some Simple Examples

    6.2 The Solution of Difference Equations by Means of Generating Functions

    6.3 Some Combinatorial Identities

    6.4 Additional Examples

    6.5 Derivatives and Differential Equations

    Part II. Existence

    7. Some Methods of Proof

    7.1 Existence by Construction

    7.2 The Method of Exhaustion

    7.3 The Dirichlet Drawer Principle

    7.4 The Method of Contradiction

    8. Geometry of the Plane

    8.1 Convex Sets

    8.2 Tiling a Rectangle

    8.3 Tessellations of the Plane

    8.4 Some Equivalence Classes

    9. Maps on a Sphere

    9.1 Euler's Formula

    9.2 Regular Maps in the Plane

    9.3 Platonic Solids

    10. Coloring Problems

    10.1 The Four Color Problem

    10.2 Coloring Graphs

    10.3 More about Chromatic Polynomials

    10.4 Chromatic Triangles

    10.5 Sperner's Lemma

    11. Finite Structures

    11.1 Finite Fields

    11.2 The Fano Plane

    11.3 Coordinate Geometry

    11.4 Projective Configurations

    Part III. Applications

    12. Probability

    12.1 Combinatorial Probability

    12.2 Ultimate Sets

    13. Ramifications of the Binomial Theorem

    13.1 Arithmetic Power Series

    13.2 The Binomial Distribution

    13.3 Distribution of Objects into Boxes

    13.4 Stirling Numbers

    13.5 Gaussian Binomial Coefficients

    14. More Generating Functions and Difference Equations

    14.1 The Partition of Integers

    14.2 Triangulation of Convex Polygons

    14.3 Random Walks

    14.4 A Class of Difference Equations

    15. Fibonacci Sequences

    15.1 Representations of Fibonacci Sequences

    15.2 Diagonal Sums of the Pascal Triangle

    15.3 Sequences of Plus and Minus Signs

    15.4 Counting Hares

    15.5 Maximum or Minimum of a Unimodal Function

    16. Arrangements

    16.1 Systems of Distinct Representatives

    16.2 Latin Squares

    16.3 The Kirkman Schoolgirl Problem

    16.4 Balanced Incomplete Block Designs

    16.5 Difference Sets

    16.6 Magic Squares

    16.7 Room Squares

    Answers to Selected Exercises

    Index

Product details

  • No. of pages: 314
  • Language: English
  • Copyright: © Academic Press 1972
  • Published: April 28, 1972
  • Imprint: Academic Press
  • eBook ISBN: 9781483273822

About the Authors

Gerald Berman

K. D. Fryer

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