Walter D. Wallis, John C. George

Introduction to Combinatorics (eBook, PDF)

• Format: PDF

Produktbeschreibung

Accessible to undergraduate students, Introduction to Combinatorics presents approaches for solving counting and structural questions. It looks at how many ways a selection or arrangement can be chosen with a specific set of properties and determines if a selection or arrangement of objects exists that has a particular set of properties.To give stu

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Produktdetails

• Verlag: Taylor & Francis eBooks
• Seitenzahl: 396
• Erscheinungstermin: 7. September 2010
• Englisch
• ISBN-13: 9781439894996
• Artikelnr.: 57150447

Autorenporträt

W.D. Wallis is Emeritus Professor of Mathematics at Southern Illinois University. His research interests include combinatorial designs, Latin squares, graph labeling, one-factorizations, and intelligent networks. Dr. Wallis is the author of Introduction to Combinatorial Designs, Second Edition (CRC Press, 2007).

J.C. George is an assistant professor of mathematics in the Division of Mathematics and Natural Sciences at Gordon College in Barnesville, Georgia. His research interests include one-factorizations, graph products, and the relationships of algebraic structures to combinatorial objects.

Inhaltsangabe

Introduction

Some Combinatorial Examples

Sets, Relations and Proof Techniques

Two Principles of Enumeration

Graphs

Systems of Distinct Representatives

Fundamentals of Enumeration

Permutations and Combinations

Applications of P(n, k) and (n k)

Permutations and Combinations of Multisets

Applications and Subtle Errors

Algorithms

Probability

Introduction

Some Definitions and Easy Examples

Events and Probabilities

Three Interesting Examples

Probability Models

Bernoulli Trials

The Probabilities in Poker

The Wild Card Poker Paradox

The Pigeonhole Principle and Ramsey's Theorem

The Pigeonhole Principle

Applications of the Pigeonhole Principle

Ramsey's Theorem - the Graphical Case

Ramsey Multiplicity

Sum-Free Sets

Bounds on Ramsey Numbers

The General Form of Ramsey's Theorem

The Principle of Inclusion and Exclusion

Unions of Events

The Principle

Combinations with Limited Repetitions

Derangements

Generating Functions and Recurrence Relations

Generating Functions

Recurrence Relations

From Generating Function to Recurrence

Exponential Generating Functions

Catalan, Bell and Stirling Numbers

Introduction

Catalan Numbers

Stirling Numbers of the Second Kind

Bell Numbers

Stirling Numbers of the First Kind

Computer Algebra and Other Electronic Systems

Symmetries and the P´olya-Redfield Method

Introduction

Basics of Groups

Permutations and Colorings

An Important Counting Theorem

P´olya and Redfield's Theorem

Partially-Ordered Sets

Introduction

Examples and Definitions

Bounds and lattices

Isomorphism and Cartesian products

Extremal set theory: Sperner's and Dilworth's theorems

Introduction to Graph Theory

Degrees

Paths and Cycles in Graphs

Maps and Graph Coloring

Further Graph Theory

Euler Walks and Circuits

Application of Euler Circuits to Mazes

Hamilton Cycles

Trees

Spanning Trees

Coding Theory

Errors; Noise

The Venn Diagram Code

Binary Codes; Weight; Distance

Linear Codes

Hamming Codes

Codes and the Hat Problem

Variable-Length Codes and Data Compression

Latin Squares

Introduction

Orthogonality

Idempotent Latin Squares

Partial Latin Squares and Subsquares

Applications

Balanced Incomplete Block Designs

Design Parameters

Fisher's Inequality

Symmetric Balanced Incomplete Block Designs

New Designs from Old

Difference Methods

Linear Alge

Rezensionen

In Introduction to Combinatorics, Wallis (emer., Southern Illinois Univ.) and George (Gordon State College) present a well-thought-out compilation of topics covering elementary combinatorics. At the beginning, the authors present a thorough background on the fundamentals of combinatorics with topics such as permutations and combinations, the pigeonhole principle, and the principle of inclusion and exclusion. Later chapters are independent of one another and can be selected based on student and instructor interests. These topics include graph theory, coding theory, and balanced incomplete block designs. At the end of each chapter, there are exercises and problems. These vary in difficulty from straightforward practice to more involved proof problems. Solutions and/or hints are provided in the back of the book. In addition, three appendixes discuss proof techniques, matrices and vectors, and historical figures; these allow flexibility in covering the material in various ways that can be based on students' backgrounds. Overall, this textbook is a highly readable work that will benefit and enlighten all those interested in learning about combinatorics. It will work in a traditional classroom setting and for independent study. Given the level of material, it is geared toward junior or senior level undergraduate students.
--S. L. Sullivan, Catawba College