How to Count An Introduction to Combinatorics 2nd Edition by Allenby, Alan Slomson 1032919779 9781032919775

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ISBN 10: 1032919779

ISBN 13: 9781032919775

Author: R.B.J.T. Allenby, Alan Slomson

Emphasizes a Problem Solving Approach A first course in combinatorics Completely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics. New to the Second Edition This second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises. Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Polya’s counting theorem.

How to Count An Introduction to Combinatorics 2nd Table of contents:

Chapter 1 What’s It All About?

1.1 What Is Combinatorics?

1.2 Classic Problems

1.3 What You Need To Know

1.4 Are You Sitting Comfortably?

Chapter 2 Permutations and Combinations

2.1 The Combinatorial Approach

2.2 Permutations

2.3 Combinations

2.4 Applications To Probability Problems

2.5 The Multinomial Theorem

2.6 Permutations and Cycles

Chapter 3 Occupancy Problems

3.1 Counting the Solutions of Equations

3.2 New Problems From Old

3.3 A “Reduction” Theorem for the Stirling Numbers

Chapter 4 The Inclusion-Exclusion Principle

4.1 Double Counting

4.2 Derangements

4.3 A Formula for the Stirling Numbers

Chapter 5 Stirling and Catalan Numbers

5.1 Stirling Numbers

5.2 Permutations and Stirling Numbers

5.3 Catalan Numbers

Chapter 6 Partitions and Dot Diagrams

6.1 Partitions

6.2 Dot Diagrams

6.3 A Bit of Speculation

Conjecture

Proof of the Conjecture

6.4 More Proofs Using Dot Diagrams

Chapter 7 Generating Functions and Recurrence Relations

7.1 Functions and Power Series

7.2 Generating Functions

7.3 What Is A Recurrence Relation?

7.4 Fibonacci Numbers

7.5 Solving Homogeneous Linear Recurrence Relations

7.6 Nonhomogeneous Linear Recurrence Relations

7.7 The Theory of Linear Recurrence Relations

7.8 Some Nonlinear Recurrence Relations

Chapter 8 Partitions and Generating Functions

8.1 The Generating Function for the Partition Numbers

8.2 A Quick(Ish) Way of Finding p(n)

8.3 An Upper Bound for the Partition Numbers

8.4 The Hardy-Ramanujan Formula

8.5 The Story of Hardy and Ramanujan

Chapter 9 Introduction to Graphs

9.1 Graphs and Pictures

9.2 Graphs: A Picture-Free Definition

9.3 Isomorphism of Graphs

9.4 Paths and Connected Graphs

9.5 Planar Graphs

9.6 Eulerian Graphs

9.7 Hamiltonian Graphs

9.8 The Four-Color Theorem

Chapter 10 Trees

10.1 What Is A Tree?

10.2 Labeled Trees

10.3 Spanning Trees and Minimal Connectors

10.4 The Shortest-Path Problem

Chapter 11 Groups of Permutations

11.1 Permutations As Groups

11.2 Symmetry Groups

11.3 Subgroups and Lagrange’S Theorem

11.4 Orders of Group Elements

11.5 The Orders of Permutations

Chapter 12 Group Actions

12.1 Colorings

12.2 The Axioms for Group Actions

12.3 Orbits

12.4 Stabilizers

Chapter 13 Counting Patterns

13.1 Frobenius’S Counting Theorem

13.2 Applications of Frobenius’S Counting Theorem

Chapter 14 Pólya Counting

14.1 Colorings and Group Actions

14.2 Pattern Inventories

14.3 The Cycle Index of A Group

14.4 PÓLya’S Counting Theorem: Statement and Examples

14.5 PÓLya’S Counting Theorem: the Proof

14.6 Counting Simple Graphs

Chapter 15 Dirichlet’s Pigeonhole Principle

15.1 The Origin of the Principle

15.2 The Pigeonhole Principle

15.3 More Applications of the Pigeonhole Principle

Chapter 16 Ramsey Theory

16.1 What Is Ramsey’S Theorem?

16.2 Three Lovely Theorems

16.3 Graphs of Many Colors

16.4 Euclidean Ramsey Theory

Chapter 17 Rook Polynomials and Matchings

17.1 How Rook Polynomials Are Defined

17.2 Matchings and Marriages

Solutions to the A Exercises

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Books for Further Reading

Mathematical Background

Combinatorics in General

Graph Theory

Group Theory

PÓLya Counting

Ramsey Theory

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