组合数学 (Fall 2011)

组合数学 Combinatorics

Instructor
尹一通
Email	yitong.yin@gmail.com yinyt@nju.edu.cn yinyt@lamda.nju.edu.cn
office	TBA
Class
Class meetings	TBA TBA
Office hours	TBA
Textbook
van Lint and Wilson, A course in Combinatorics, 2nd Ed, Cambridge Univ Press, 2001.

This is the page for the class Combinatorics for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.

Course info

• Instructor : 尹一通

• email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
• office:

• Teaching fellow:

• email:

• Class meeting:
• Office hour:

Syllabus

先修课程 Prerequisites

• 离散数学（Discrete Mathematics）
• 线性代数（Linear Algebra）
• 概率论（Probability Theory）

Course materials

• 教材和参考书清单

成绩 Grades

• 课程成绩：本课程将会有六次作业和一次期末考试。最终成绩将由平时作业成绩和期末考试成绩综合得出。
• 迟交：如果有特殊的理由，无法按时完成作业，请提前联系授课老师，给出正当理由。否则迟交的作业将不被接受。

学术诚信 Academic Integrity

学术诚信是所有从事学术活动的学生和学者最基本的职业道德底线，本课程将不遗余力的维护学术诚信规范，违反这一底线的行为将不会被容忍。

作业完成的原则：署你名字的工作必须由你完成。允许讨论，但作业必须独立完成，并在作业中列出所有参与讨论的人。不允许其他任何形式的合作——尤其是与已经完成作业的同学“讨论”。

本课程将对剽窃行为采取零容忍的态度。在完成作业过程中，对他人工作（出版物、互联网资料、其他人的作业等）直接的文本抄袭和对关键思想、关键元素的抄袭，按照 ACM Policy on Plagiarism的解释，都将视为剽窃。剽窃者成绩将被取消。如果发现互相抄袭行为， 抄袭和被抄袭双方的成绩都将被取消。因此请主动防止自己的作业被他人抄袭。

学术诚信影响学生个人的品行，也关乎整个教育系统的正常运转。为了一点分数而做出学术不端的行为，不仅使自己沦为一个欺骗者，也使他人的诚实努力失去意义。让我们一起努力维护一个诚信的环境。

Lecture Notes

1. Basic enumeration
2. Partitions, Sieve methods
3. Generating functions
4. Existence, the probabilistic method
5. Random graphs
6. Extremal graphs
7. Finite set systems
8. Flow and matching
9. Extremal set theory
10. Extremal set theory II
11. Ramsey theory
12. The Szemeredi regularity lemma
13. Optimization
14. Duality, Matroid

Concepts

• Binomial coefficient
• Composition of a number
• Combinations with repetition, [math]\displaystyle{ k }[/math]-multisets on a set
• Stirling number of the second kind
• Partition of a number
• The twelvefold way
• Ferrers diagram (and the MathWorld link)
• The principle of inclusion-exclusion (and more generally the sieve method)
• Derangement, and Problème des ménages
• Generating function and formal power series
• Fibonacci number
• Newton's formula
• Catalan number
• Double counting and the handshaking lemma
• Cayley's formula
• Pigeonhole principle

• Erdős–Szekeres theorem
• Dirichlet's approximation theorem

• The Probabilistic Method
• Erdős–Rényi model for random graphs
• Graph property
• Some graph parameters: girth [math]\displaystyle{ g(G) }[/math], chromatic number [math]\displaystyle{ \chi(G) }[/math], Independence number [math]\displaystyle{ \alpha(G) }[/math], clique number [math]\displaystyle{ \omega(G) }[/math]
• Extremal graph theory
• Turán's theorem, Turán graph
• Two analytic inequalities:

• Cauchy–Schwarz inequality
• the inequality of arithmetic and geometric means

• Erdős–Stone theorem (fundamental theorem of extremal graph theory)
• Dirac's theorem
• Hall's theorem (the marriage theorem)
• Birkhoff-Von Neumann theorem
• König-Egerváry theorem
• Dilworth's theorem
• Sperner system
• Erdős–Ko–Rado theorem
• VC dimension
• Kruskal–Katona theorem
• Ramsey theory

• Ramsey's theorem
• Happy Ending problem
• Van der Waerden's theorem
• Hales–Jewett theorem

• Lovász local lemma
• Combinatorial optimization

• optimization
• convex combination, convex set, convex function
• local optimum (see also maxima and minima)

• Linear programming

• linear constraint
• hyperplane, halfspace, polyhedron, convex polytope
• the Simplex algorithm

• The Max-Flow Min-Cut Theorem

• Maximum flow
• minimum cut

• Unimodularity
• Duality

• LP Duality

• Matroid

• weighted matroid and greedy algorithm
• Matroid intersection

• Laplacian
• [math]\displaystyle{ \lambda_2 }[/math] of a graph and Cheeger Inequalities
• Expander graph
• Szemerédi regularity lemma