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

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. Extremal set theory
9. Extremal set theory II
10. Ramsey theory
11. Optimization
12. Flow and matching
13. Duality, Matroid
14. Graph spectrum, expanders
15. The Szemeredi regularity lemma

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