- Page 4 and 5: Combinatorics Through Guided Discov
- Page 6 and 7: About the Author Kenneth Paul Bogar
- Page 8 and 9: Preface This book is an introductio
- Page 10 and 11: ix will find a real sense of satisf
- Page 12 and 13: Preface to PreTeXt edition At the t
- Page 14 and 15: Contents About the Author Preface P
- Page 16 and 17: Contents xv 5 The Principle of Incl
- Page 18 and 19: Chapter 1 What is Combinatorics? Co
- Page 20 and 21: 1.2. Basic Counting Principles 3 te
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- Page 24 and 25: 1.2. Basic Counting Principles 7 +
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- Page 38 and 39: 1.3. Some Applications of the Basic
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- Page 46 and 47: 1.4. Supplementary Chapter Problems
- Page 48 and 49: Chapter 2 Applications of Induction
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2.1. Some Examples of Mathematical
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2.2. Recurrence Relations 37 ⇒ Pr
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2.2. Recurrence Relations 39 Figure
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2.3. Graphs and Trees 41 Problem 99
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2.3. Graphs and Trees 43 (c) Find a
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2.3. Graphs and Trees 45 (c) Can yo
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2.3. Graphs and Trees 47 2.3.6 The
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2.4. Supplementary Problems 49 edge
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Chapter 3 Distribution Problems 3.1
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3.1. The idea of a distribution 53
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3.1. The idea of a distribution 55
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3.1. The idea of a distribution 57
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3.2. Partitions and Stirling Number
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3.2. Partitions and Stirling Number
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3.3. Partitions of Integers 63 (d)
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3.3. Partitions of Integers 65 •
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3.3. Partitions of Integers 67 With
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3.3. Partitions of Integers 69 The
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3.4. Supplementary Problems 71 (a)
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Chapter 4 Generating Functions 4.1
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4.1. The Idea of Generating Functio
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4.1. The Idea of Generating Functio
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4.1. The Idea of Generating Functio
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4.2. Generating functions for integ
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4.2. Generating functions for integ
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4.3. Generating Functions and Recur
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4.3. Generating Functions and Recur
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4.3. Generating Functions and Recur
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4.4. Supplementary Problems 91 Writ
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Chapter 5 The Principle of Inclusio
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5.1. The size of a union of sets 95
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5.2. Application of Inclusion and E
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5.3. Deletion-Contraction and the C
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5.4. Supplementary Problems 101 tre
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Chapter 6 Groups acting on sets 6.1
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6.1. Permutation Groups 105 What yo
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6.1. Permutation Groups 107 and σ(
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6.1. Permutation Groups 109 Problem
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6.1. Permutation Groups 111 line) a
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6.1. Permutation Groups 113 Problem
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6.2. Groups Acting on Sets 115 We h
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6.2. Groups Acting on Sets 117 1 2
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6.2. Groups Acting on Sets 119 6.2.
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6.2. Groups Acting on Sets 121 Prob
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6.2. Groups Acting on Sets 123 Prob
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6.3. Pólya-Redfield Enumeration Th
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6.3. Pólya-Redfield Enumeration Th
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6.3. Pólya-Redfield Enumeration Th
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6.4. Supplementary Problems 131 ⇒
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Appendix A Relations A.1 Relations
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A.1. Relations as sets of Ordered P
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A.1. Relations as sets of Ordered P
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A.2. Equivalence relations 139 ⇒
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A.2. Equivalence relations 141 Prob
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A.2. Equivalence relations 143 Prob
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Appendix B Mathematical Induction B
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B.1. The Principle of Mathematical
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B.1. The Principle of Mathematical
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Appendix C Exponential Generating F
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C.2. Exponential Generating Functio
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C.3. Applications to recurrences. 1
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C.4. The Product Principle for EGFs
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C.4. The Product Principle for EGFs
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C.4. The Product Principle for EGFs
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C.5. The Exponential Formula 163 an
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C.5. The Exponential Formula 165 On
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C.6. Supplementary Problems 167 6.
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Appendix D Hints to Selected Proble
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171 37. Ask yourself “What is a p
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173 goes back to 0, up to 2 again,
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175 79. We didn’t explicitly say
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177 to the sum of the degrees. Anot
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179 identical ones. If we permute t
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181 162. How can you start with a p
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183 from the row above. Even in thi
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185 207. In the power series ∑
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187 intersections of sets. Keep on
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189 286. Before you try to show tha
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191 391. Notice that any permutatio
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Appendix E GNU Free Documentation L
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195 begin distribution of Opaque co
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197 of those notices and disclaimer
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Index S(k, n), 58 Π notation, 8 n!
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Index 201 geometric progression, 40
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Index 203 sequence degree, 50 serie