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Combinatorics Math 3U03: Combinatorics Jeremy Lane December 3, 2020 Contents 1 Introduction to the course6 1.1 Land Acknowledgement.......................6 1.2 Welcome to the course........................6 1.3 Organization.............................7 1.4 What is Combinatorics?.......................7 1.5 Practice Problems..........................9 2 Sets and Counting 11 2.1 Sets.................................. 11 2.2 Counting............................... 11 2.3 Example: Tilings of a 1 × n chessboard.............. 12 2.4 Example: A combinatorial proof of an identity about tilings... 13 2.5 Practice Problems.......................... 14 3 Counting with bijections 16 3.1 Functions............................... 16 3.2 Injective, Surjective, Bijective.................... 16 3.3 Counting sets using functions.................... 17 3.4 Example: words in the alphabet fa; bg ............... 18 3.5 Practice problems........................... 18 4 Induction and finite disjoint unions 20 4.1 Induction............................... 20 4.2 Disjoint unions............................ 22 4.3 Another domino example...................... 22 4.4 Practice problems........................... 23 5 Cartesian products and multiplication 26 5.1 Cartesian product.......................... 26 5.2 The multiplication principle..................... 26 5.3 Practice problems........................... 28 1 6 Permutations 29 6.1 Permutations and non-attacking rooks............... 29 6.2 Notation for permutations...................... 30 6.3 Counting Permutations....................... 30 6.4 Practice problems........................... 31 7 Proofs 32 7.1 Before writing a proof........................ 32 7.2 Early stages of writing a proof................... 32 7.3 Mid stages of writing a proof.................... 33 7.4 What if I get stuck?......................... 33 7.5 Late stages of writing a proof.................... 33 7.6 Sanity checks............................. 34 7.7 Reading and understanding a proof................. 34 7.8 Evaluating a written proof...................... 35 7.9 Proof by induction: some additional techniques.......... 35 7.10 Practice problems........................... 37 8 Ordered subsets of [n] 39 8.1 Permutations of [n].......................... 39 8.2 Seating problems........................... 39 8.3 Ordered subsets of [n]........................ 39 8.4 An identity for n place k ....................... 40 8.5 Practice problems........................... 41 9 Cycles and Seatings 42 9.1 Cycles of a permutation....................... 42 9.2 Cycle notation............................ 42 9.3 Examples............................... 43 9.4 Seating problems........................... 44 9.5 Practice problems........................... 46 10 Stirling numbers of the first kind 47 10.1 Examples............................... 47 10.2 Stirling numbers of the first kind.................. 49 10.3 An identity for Stirling numbers of the first kind......... 50 10.4 Practice problems........................... 50 11 Binomial coefficients, part 1 51 11.1 Subsets................................ 51 11.2 Binomial coefficients......................... 51 11.3 An identity relating binomial coefficients and tiling numbers... 52 11.4 Practice problems........................... 52 2 12 Binomial coefficients, part 2 54 12.1 A generalization of a familiar identity............... 54 12.2 Pascal's recurrence.......................... 55 12.3 Binomial theorem........................... 55 12.4 Practice problems........................... 56 13 Binomial coefficients, part 3 57 13.1 Vandermonde identity........................ 57 13.2 Poker................................. 57 13.3 Stars and bars............................ 58 13.4 A distribution problem........................ 58 13.5 Lattice paths............................. 59 13.6 Catalan numbers........................... 59 13.7 Practice problems........................... 60 14 Multinomial coefficients 61 14.1 Example: distributing candies.................... 61 14.2 Multinomial coefficients....................... 62 14.3 Generalized stars and bars...................... 63 14.4 Pascal's recurrence.......................... 65 14.5 The multinomial theorem...................... 66 14.6 Practice problems........................... 67 15 Principle of Inclusion and Exclusion 68 15.1 Principle of Inclusion and Exclusion, version 1........... 68 15.2 Principle of Inclusion and Exclusion, version 2........... 69 15.3 Principle of Inclusion and Exclusion, version 3........... 70 15.4 practice problems........................... 71 16 More combinatorial numbers (and related distribution prob- lems) 72 16.1 Stirling numbers of the second kind................. 72 16.2 Bell numbers............................. 72 16.3 Partition numbers.......................... 73 16.4 Practice problems........................... 74 17 Derangements and the gift exchange problem 75 17.1 The gift exchange problem...................... 75 17.2 Derangements............................. 75 17.3 Counting derangements....................... 75 17.4 Practice problems........................... 76 18 Introduction to generating functions 77 18.1 Term paper.............................. 77 18.2 Introduction to generating functions................ 77 18.3 Practice problems........................... 79 3 19 Power series 80 19.1 Formal power series......................... 80 19.2 Power series.............................. 81 19.3 Examples............................... 82 19.4 Practice problems........................... 84 19.5 Bonus practice problems....................... 84 20 Products of generating functions 86 20.1 Multiplying functions and power series............... 86 20.2 Proving identities by multiplying generating functions...... 86 20.3 The convolution principle...................... 87 20.4 Practice problems........................... 88 21 Applications of generating functions 90 21.1 Constructing generating functions for counting problems..... 90 21.2 The method of generating functions................ 90 21.3 Practice problems........................... 90 22 Exponential generating functions 91 22.1 Exponential generating functions.................. 91 22.2 Rules for exponential generating functions............. 91 22.3 Example: Bell numbers....................... 92 22.4 Practice problems........................... 93 23 Solving linear recurrence relations 94 23.1 Classifying recurrence relations................... 94 23.2 The method of characteristic polynomials............. 94 23.3 Why it works............................. 95 23.4 Practice problems........................... 96 24 Existence and proof by contradiction 97 24.1 Existence problems in combinatorics................ 97 24.2 Example: perfect covers....................... 97 24.3 Proof by contradiction........................ 98 24.4 Practice problems........................... 99 25 The pigeonhole principle 100 25.1 The pigeonhole principle....................... 100 25.2 Practice problems........................... 100 26 Incidence structures and block designs 101 26.1 Incidence structures......................... 101 26.2 Regular block designs........................ 101 27 Balanced Incomplete Block Designs (BIBDs) 103 27.1 Practice problems........................... 104 4 28 Steiner Triple Systems 105 28.1 Kirkman's schoolgirls......................... 105 28.2 Steiner Triple Systems........................ 105 28.3 Practice problems........................... 106 29 Finite projective planes, part I 109 29.1 Finite projective planes....................... 109 29.2 Practice problems........................... 109 30 Finite projective planes, part II 110 30.1 Finite projective planes....................... 110 30.2 Practice problems........................... 111 A Sets 112 5 1 Introduction to the course • Date: September 8, 2020 • Reading: { How To Count, Section 1.1. • To-do: { Complete the questionnaire on Avenue about Math Experience by the end of Wednesday. { Introduce yourself in the Avenue discussion. 1.1 Land Acknowledgement We acknowledge the traditional territories upon which we gather, whether vir- tually or in person. McMaster University is located on the traditional territories of the Mississauga and Haudenosaunee nations, and within the lands protected by the \Dish with One Spoon" wampum agreement. 1.2 Welcome to the course • Please take a minute to briefly introduce yourself in the Avenue discussion \Welcome to Math 3U03!" • Please complete the survey on Avenue about Math Experience by the end of Wednesday. This will help me to better calibrate the beginning of the course to everyone's benefit. • Especially in this semester, where things are going to be a bit more com- plicated than normal, I want to ensure everyone is getting the support they need. I will do my best to provide you with the tools to succeed in the course. If anyone finds they are struggling and need any additional support, please let me know. • It is a good time to remember that there are various university resources available, such as: { Student Accessibility Services. { University Technology Services. { Student Support Services (which includes wellness and mental health resources). { Math Help Centre. 6 1.3 Organization • Textbook: { We will be using \How to Count" by Beeler. { Pdf copies are available through the library. • Avenue to Learn: { Lectures, practice problems, and assignments will all be posted to Avenue. { We also have a discussion board. • Zoom. { Please keep your microphone muted
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