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Combinatorics an Upper-Level Introductory Course in Enumeration, Graph Theory, and Design Theory

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Combinatorics an Upper-Level Introductory Course in Enumeration, Graph Theory, and Design Theory Combinatorics an upper-level introductory course in enumeration, graph theory, and design theory by Joy Morris University of Lethbridge Version 2.0 of July 2021 This book is offered under the Creative Commons license. (Attribution-NonCommercial-ShareAlike 2.0) Please send comments and corrections to: [email protected] © 2014–2021 by Joy Morris. Some rights reserved. You are free to copy this book, to distribute it, to display it, and to make derivative works, under the following conditions: (1) Attribution. You must give the original author credit. (2) Noncommercial. You may not use this work for commercial purposes. (3) Share Alike. If you alter, transform, or build upon this work, you may distribute the resulting work only under a license identical to this one. — For any reuse or distribution, you must make clear to others the license terms of this work. Any of these conditions can be waived if you get permission from the copyright holder. Your fair use and other rights are in no way affected by the above. — This is a human-readable summary of the full license, whichis available on-line at http://creativecommons.org/licenses/by-nc-sa/2.0/legalcode Contents Chapter 1. What is Combinatorics? 1 §1.1. Enumeration ................................................................. 1 §1.2. Graph Theory -------------------------------------------------------- 2 §1.3. Ramsey Theory .............................................................. 3 §1.4. Design Theory -------------------------------------------------------- 4 §1.5. Coding Theory ............................................................... 4 Part I. Enumeration Chapter 2. Basic Counting Techniques 9 §2.1. The product rule ............................................................. 9 §2.2. The sum rule -------------------------------------------------------- 12 §2.3. Putting them together .......................................................14 §2.4. Summing up --------------------------------------------------------- 17 Chapter 3. Permutations, Combinations, and the Binomial Theorem 19 §3.1. Permutations ................................................................19 §3.2. Combinations -------------------------------------------------------- 22 §3.3. The Binomial Theorem ......................................................26 Chapter 4. Bijections and Combinatorial Proofs 29 §4.1. Counting via bijections ......................................................29 §4.2. Combinatorial proofs ------------------------------------------------- 31 §4.3. The Arithmetic Triangle (Pascal’s Triangle) ..................................37 Chapter 5. Counting with Repetitions 39 §5.1. Unlimited repetition .........................................................39 §5.2. Sorting a set that contains repetition ----------------------------------- 42 Chapter 6. Induction and Recursion 47 §6.1. Recursively-defined sequences ................................................47 §6.2. Basic induction ------------------------------------------------------ 49 §6.3. More advanced induction ....................................................52 Chapter 7. Generating Functions 59 §7.1. What is a generating function? ..............................................59 §7.2. The Generalised Binomial Theorem ------------------------------------- 60 §7.3. Using generating functions to count things ...................................62 i ii Chapter 8. Generating Functions and Recursion 67 §8.1. Partial fractions .............................................................67 §8.2. Factoring polynomials ------------------------------------------------ 69 §8.3. Using generating functions to solve recursively-defined sequences .............71 Chapter 9. Some Important Recursively-Defined Sequences 79 §9.1. Derangements ...............................................................79 §9.2. Catalan numbers ----------------------------------------------------- 80 §9.3. Bell numbers and exponential generating functions ...........................83 Chapter 10. Other Basic Counting Techniques 89 §10.1. The Pigeonhole Principle ....................................................89 §10.2. Inclusion-Exclusion --------------------------------------------------- 94 Part II. Graph Theory Chapter 11. Basics of Graph Theory 103 §11.1. Background ................................................................103 §11.2. Basic definitions, terminology, and notation ----------------------------- 104 §11.3. Subgraphs, complete graphs, and the Handshaking Lemma ..................107 §11.4. Isomorphism of graphs ----------------------------------------------- 110 §11.5. Random graphs ............................................................115 Chapter 12. Moving through graphs 119 §12.1. Directed graphs ............................................................119 §12.2. Walks and connectedness --------------------------------------------- 120 §12.3. Paths and cycles ...........................................................123 §12.4. Trees -------------------------------------------------------------- 125 §12.5. Automorphisms of graphs ...................................................128 Chapter 13. Euler and Hamilton 133 §13.1. Euler tours and trails .......................................................133 §13.2. Hamilton paths and cycles -------------------------------------------- 137 Chapter 14. Graph Colouring 143 §14.1. Edge colouring .............................................................143 §14.2. Ramsey Theory ----------------------------------------------------- 149 §14.3. Vertex colouring ............................................................154 Chapter 15. Planar graphs 161 §15.1. Planar graphs ..............................................................161 §15.2. Euler’s Formula ----------------------------------------------------- 167 §15.3. Map colouring ..............................................................171 iii Part III. Design Theory Chapter 16. Latin squares 179 §16.1. Latin squares and Sudokus .................................................179 §16.2. Mutually orthogonal Latin squares (MOLS) ---------------------------- 181 §16.3. Systems of distinct representatives ..........................................186 Chapter 17. Designs 193 §17.1. Balanced Incomplete Block Designs (BIBD) .................................193 §17.2. Constructing designs, and existence of designs - - - - - - - - - - - - - - - - - - - - - - - - - - 197 §17.3. Fisher’s Inequality ..........................................................201 Chapter 18. More designs 205 §18.1. Steiner and Kirkman triple systems .........................................205 §18.2. t-designs ----------------------------------------------------------- 210 §18.3. Affine planes ...............................................................213 §18.4. Projective planes ---------------------------------------------------- 219 Chapter 19. Designs and Codes 221 §19.1. Introduction ................................................................221 §19.2. Error-correcting codes ----------------------------------------------- 222 §19.3. Using the generator matrix for encoding ....................................225 §19.4. Using the parity-check matrix for decoding ----------------------------- 227 §19.5. Codes from designs .........................................................232 Appendix A. Complex Numbers 235 Appendix B. Biographical Briefs 241 B.1. List of Entries ...............................................................242 B.2. Biographies ---------------------------------------------------------- 244 iv Appendix C. Solutions to selected exercises 305 Solutions for Chapter 2 ......................................................305 Solutions for Chapter 3 ----------------------------------------------- 306 Solutions for Chapter 4 ......................................................307 Solutions for Chapter 5 ----------------------------------------------- 309 Solutions for Chapter 6 ......................................................310 Solutions for Chapter 7 ----------------------------------------------- 311 Solutions for Chapter 8 ......................................................313 Solutions for Chapter 9 ----------------------------------------------- 317 Solutions for Chapter 10 .....................................................318 Solutions for Chapter 11 ---------------------------------------------- 319 Solutions for Chapter 12 .....................................................321 Solutions for Chapter 13 ---------------------------------------------- 323 Solutions for Chapter 14 .....................................................324 Solutions for Chapter 15 ---------------------------------------------- 327 Solutions for Chapter 16 .....................................................329 Solutions for Chapter 17 ---------------------------------------------- 330 Solutions for Chapter 18 .....................................................332 Solutions for Chapter 19 ---------------------------------------------- 337 Appendix D. List of Notation 341 Index 343 Chapter 1 What is Combinatorics? Combinatorics is a subfield of “discrete mathematics,” so we should begin by asking what discrete mathematics means. The differences are to some extent a matter of opinion, and various mathematicians might classify specific topics differently. “Discrete” should not be confused with “discreet,” which is a much more commonly-used word. They share the same Latin root, “discretio,” which
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