Combinatorics (Fall 2010)

组合数学 Combinatorics

Instructor
尹一通
Email	yitong.yin@gmail.com yinyt@nju.edu.cn yinyt@lamda.nju.edu.cn
office	蒙民伟楼 406
Class
Class meetings	10am -12am, Friday, 馆I-105
Office hours	2pm-5pm, Saturday, MMW 406
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 2010 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

• (2010/10/02) 第四次课讲义已补完, 讲义最后附有参考阅读的章节.
• (2010/09/27) 第一次交作业的名单已公布，见assignment。如果交了作业但没有看到名字，请与我联系。
• (2010/09/24) 提醒：明天（9月25日）上课时交作业。
• (2010/09/15) 第一次作业已发布，9月25日课上交。
• (2010/09/15) 第二课的lecture notes和slides中有个错误，

[math]\displaystyle{ |U|-\sum_{I\subseteq\{1,\ldots,n\}}(-1)^{|I|}\left|\bigcap_{i\in I}A_i\right| }[/math] 的正负号有误，应为 [math]\displaystyle{ |U|+\sum_{I\subseteq\{1,\ldots,n\}}(-1)^{|I|}\left|\bigcap_{i\in I}A_i\right| }[/math] 或 [math]\displaystyle{ |U|-\sum_{I\subseteq\{1,\ldots,n\}}(-1)^{|I|-1}\left|\bigcap_{i\in I}A_i\right| }[/math]。

感谢林文敏同学发现这个错误。

• (2010/09/11) 前两节课slides已上传，在lecture notes的部分有链接。
• 9月3日第一次课。时间：上午三、四节；地点：馆I-105。

Course info

• Instructor : 尹一通

• email: yitong.yin@gmail.com, yinyt@nju.edu.cn, yinyt@lamda.nju.edu.cn
• office: MMW 406.

• Teaching fellow: 林木丰

• email: forest.sky.sea@gmail.com

• Class meeting: 10am-12 am, Friday; 馆I-105.
• Office hour: 2-5pm, Saturday; MMW 406.

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

Policies

• Policies

Assignments

• (2010/09/17) Problem set 1 due on Sept 25, in class.

交作业名单

• (2010/09/27) 第一次交作业名单

Lecture Notes

A tentative list of topics:

1. Basic enumeration | slides
2. Partitions, Sieve methods | slides
3. Generating functions | slides
4. Existence, the probabilistic method | slides
5. Random graphs | slides
6. Extremal graph theory
7. Finite set systems
8. Extremal set theory
9. Ramsey theory
10. Optimization
11. Duality
12. Flow and matching
13. Matroid
14. Spectra of graphs
15. Harmonic analysis of boolean functions
16. The Szemeredi regularity lemma
17. Sum-product theorems, Kakeya set

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]