vi Contents Chapter 3. Integration in Asymptopia 37 §3.1. The Gaussian Tail 38 §3.2. High Trigonometric Powers 40 §3.3. An Easy Integral 43 §3.4. Integrals with logs 44 Chapter 4. From Integrals to Sums 47 §4.1. Approximating Sums by Integrals 48 §4.2. The Harmonic Numbers 51 Chapter 5. Asymptotics of Binomial Coefficients ( n k ) 57 §5.1. k Relatively Small 57 §5.2. Some Exercises 60 §5.3. k Linear in n 63 §5.4. At and Near the Middle Binomial Coefficient 66 §5.5. The Binomial Distribution 68 §5.6. The Binomial Tail 68 Chapter 6. Unicyclic Graphs 71 §6.1. Rooted Trees 72 §6.2. Rooted Trees to Pr¨ ufer Sequences 73 §6.3. Pr¨ ufer Sequences to Rooted Trees 79 §6.4. Proof of Bijection 82 §6.5. Rooted Forests 83 §6.6. Pr¨ ufer Sequences to Rooted Forests 83 §6.7. . . . and Back Again 86 §6.8. An Exact Formula for Unicyclic Graphs 88 §6.9. Counting Unicyclic Graphs in Asymptopia 90 Chapter 7. Ramsey Numbers 93 §7.1. Initial Erd˝ os Argument 93 §7.2. Deletion 94 §7.3. Lov´ asz Local Lemma 95
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