组合数学 (Spring 2013): Difference between revisions

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* [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma]
* [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma]
* [http://en.wikipedia.org/wiki/Sperner's_lemma Sperner's lemma] and [http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem Brouwer fixed point theorem]
* [http://en.wikipedia.org/wiki/Sperner's_lemma Sperner's lemma] and [http://en.wikipedia.org/wiki/Brouwer_fixed_point_theorem Brouwer fixed point theorem]
* [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]
* [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle]
* [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle]
:* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
:* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
:* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem]
:* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem]
* [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method]
* [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method]
* [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs]
* [http://en.wikipedia.org/wiki/Graph_property Graph property]
* Some graph parameters: [http://en.wikipedia.org/wiki/Girth_(graph_theory) girth <math>g(G)</math>], [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number <math>\chi(G)</math>], [http://mathworld.wolfram.com/IndependenceNumber.html Independence number <math>\alpha(G)</math>], [http://mathworld.wolfram.com/CliqueNumber.html clique number <math>\omega(G)</math>]
* [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory]
* [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory]
* [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph]
* [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph]
Line 153: Line 151:
:* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means]
:* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory)
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory)
* [http://en.wikipedia.org/wiki/Dirac's_theorem Dirac's theorem]
* [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem)
* [http://en.wikipedia.org/wiki/Birkhoff-Von_Neumann_theorem Birkhoff-Von Neumann theorem]
* [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem]
* [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem]
* [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
* [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
* [http://en.wikipedia.org/wiki/VC_dimension VC dimension]
* [http://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem Kruskal–Katona theorem]
* [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory]
* [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory]
:*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem]
:*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem]
:*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem]
:*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem]
:*[http://en.wikipedia.org/wiki/Van_der_Waerden's_theorem Van der Waerden's theorem]
* [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem)
:*[http://en.wikipedia.org/wiki/Hales-Jewett_theorem Hales–Jewett theorem]
* [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem]
* [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma]
* [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem]
* [http://en.wikipedia.org/wiki/Combinatorial_optimization Combinatorial optimization]
* The  [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
:* [http://en.wikipedia.org/wiki/Optimization_(mathematics) optimization]
:* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]
:* [http://en.wikipedia.org/wiki/Convex_combination convex combination], [http://en.wikipedia.org/wiki/Convex_set convex set], [http://en.wikipedia.org/wiki/Convex_function convex function]
:* [http://en.wikipedia.org/wiki/Local_optimum local optimum] (see also [http://en.wikipedia.org/wiki/Maxima_and_minima maxima and minima])
* [http://en.wikipedia.org/wiki/Linear_programming Linear programming]
* [http://en.wikipedia.org/wiki/Linear_programming Linear programming]
:* [http://en.wikipedia.org/wiki/Linear_inequality linear constraint]
:* [http://en.wikipedia.org/wiki/Hyperplane hyperplane], [http://en.wikipedia.org/wiki/Half_space halfspace], [http://en.wikipedia.org/wiki/Polyhedron polyhedron], [http://en.wikipedia.org/wiki/Convex_polytope convex polytope]
:* [http://en.wikipedia.org/wiki/Simplex_algorithm the Simplex algorithm]
*  The  [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
:* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]
:* [http://en.wikipedia.org/wiki/Minimum_cut minimum cut]
* [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity]
* [http://en.wikipedia.org/wiki/Dual_linear_program Duality]
* [http://en.wikipedia.org/wiki/Dual_linear_program Duality]
:* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality]
:* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality]
* [http://en.wikipedia.org/wiki/Matroid Matroid]
* [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity]
:* [http://en.wikipedia.org/wiki/Weighted_matroid weighted matroid] and [http://en.wikipedia.org/wiki/Greedy_algorithm greedy algorithm]
:* [http://en.wikipedia.org/wiki/Matroid_intersection Matroid intersection]
* [http://en.wikipedia.org/wiki/Laplacian_matrix Laplacian]
* [http://en.wikipedia.org/wiki/Algebraic_connectivity <math>\lambda_2</math> of a graph] and [http://en.wikipedia.org/wiki/Expander_graph#Cheeger_Inequalities Cheeger Inequalities]
* [http://en.wikipedia.org/wiki/Expander_graph Expander graph]
* [http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma Szemerédi regularity lemma]

Revision as of 14:34, 19 June 2013

组合数学
Combinatorics
Instructor
尹一通
Email yitong.yin@gmail.com yinyt@nju.edu.cn
office 计算机系 804
Class
Class meetings Thursday, 8am-10am
仙 I-103
Office hours Wednesday, 2-4pm
计算机系 804
Textbook
van Lint and Wilson.
A course in Combinatorics, 2nd ed.,
Cambridge Univ Press, 2001.
Jukna. Extremal Combinatorics:
With Applications in Computer Science,
2nd ed., Springer, 2011.
v · d · e

This is the page for the class Combinatorics for the Spring 2013 semester. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

Course info

  • Instructor : 尹一通
  • email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
  • office: 804
  • Teaching fellow: 金宇
  • email: jinyu1122@hotmail.com
  • Class meeting: Thursday 8am-10am, 仙 I-103.
  • Office hour: Wednesday 2-4pm, 计算机系 804.

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

成绩 Grades

  • 课程成绩：本课程将会有若干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。

Assignments

Lecture Notes

  1. Basic enumeration | slides1 | slides2
  2. Generating functions | slides1 | slides2
  3. Sieve methods | slides1 | slides2
  4. Cayley's formula | slides
  5. Pólya's theory of counting | slides1 | slides2
  6. Existence problems | slides1
  7. The probabilistic method
  8. Extremal graph theory
  9. Extremal set theory
  10. Ramsey theory
  11. Matching theory
  12. Flow and matching

Concepts

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