Combinatorics Lecture Notes.pdf - Introduction to Combinatorics University of Toronto Scarborough Lecture Notes Stefanos Aretakis March 4 2019 Contents

Combinatorics Lecture Notes.pdf - Introduction to...

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Introduction to Combinatorics University of Toronto Scarborough Lecture Notes Stefanos Aretakis March 4, 2019 Contents 1 Introduction 2 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The prisoners’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Solution to the prisoners’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Combinatorial Principles: Contradiction, Reduction and Induction . . . . . . . 5 2 The Pigeonhole Principle 5 2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Systematic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Ramsey Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Erd¨ os–Szekeres Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Diophantine Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 The Principle of Extremals 19 3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 The Principle of Invariants 23 4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Semi-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Permutations and Combinations 29 5.1 Additive and Multiplicative Principle . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.4 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1
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5.5 Identities of the binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . 33 5.6 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Combinations with Repetition 35 6.1 Solutions to linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 The Path Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.3 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7 Inclusion–Exclusion principle 39 7.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.3 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8 Recurrence Relations 41 8.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.3 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 9 Generating Functions 48 9.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.3 Applications in recurrence relations . . . . . . . . . . . . . . . . . . . . . . . . . 58 9.4 Practice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10 Partitions of Natural Numbers 61 10.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.2 Partitions with restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.3 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1 Introduction 1.1 Introduction Combinatotics is about... ... counting without really counting all possible cases one by one . More broadly: Combinatorics is about... ... derivining properties of structures satisfying given conditions without analyzing each and every possible case separately . For example, it is clearly possible to compute the sum 1 + 2 + · · · + 100 by adding all numbers but this is clearly inefficient and time consuming. On the other hand, if we combine the numbers in a different way (1 + 100) + (2 + 99) + · · · (50 + 51) then we can immediately conclude that the sum is equal to 101*50 (since there are 50 pairs the sum of each of which is 101). Analyzing, deriving and counting common properties of structures satisfying given con- ditions can in principle be quite challenging and require a non trivial amount of focus and concentration. This is the challenging part of our course. 2
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The fun part of our course is that the structures we will be considering are very elementary (no involved defitions, symbols etc.). Hence, for most of the time we will not be obstracted by general and abstract nonsense. In other words, we should view our course as a fun and challenging way to learn how to learn. (Question: what is the Greek word for ”learning how to learn”: Answer: Mathematics!). 1.2 The prisoners’ Problem Let’s consider the so-called ”prisoners’ problem” as a way to see a few Combinatorial principles in action: We consider an island full of male prisoners such that the following conditions hold: 1. there are 100 prisoners, 2. all have green eyes, 3. they can all see that all other prisoners have green eyes, 4. they, however, do not know that they themselves have green eyes, 5. no communication is allowed between the prisoners,
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