组合数学 (Fall 2017): Difference between revisions

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* (2017/9/4) 新学期第一次上课。
* (2017/9/4) 新学期第一次上课。
* (2017/9/18) 第一次作业发布。10月9日课上交。
* (2017/9/18) 第一次作业发布。10月9日课上交。
* (2017/10/16) 第一次作业发布。10月30日课上交。
* (2017/12/27) <font color=red size=4>考前习题讲解与作业答疑。1月6日下午2点计算机系224。</font>
* (2017/12/27) <font color=red size=4>期末考试定于1月7日下午2点整准时开始,地点在仙2-504。</font>


= Course info =
= Course info =
Line 95: Line 98:
= Assignments =
= Assignments =
*[[组合数学 (Fall 2017)/Problem Set 1|Problem Set 1]]  due on Oct 9, in class.
*[[组合数学 (Fall 2017)/Problem Set 1|Problem Set 1]]  due on Oct 9, in class.
*[[组合数学 (Fall 2017)/Problem Set 2|Problem Set 2]]  due on Oct 30, in class.
*[[组合数学 (Fall 2017)/Problem Set 3|Problem Set 3]]  due on Nov 27, in class.
*[[组合数学 (Fall 2017)/Problem Set 4|Problem Set 4]]  due on Dec 25, in class.


= Lecture Notes =
= Lecture Notes =
# [[组合数学 (Fall 2017)/Basic enumeration|Basic enumeration | 基本计数]] ( [http://tcs.nju.edu.cn/slides/comb2017/BasicEnumeration.pdf slides])
# [[组合数学 (Fall 2017)/Basic enumeration|Basic enumeration | 基本计数]] ( [http://tcs.nju.edu.cn/slides/comb2017/BasicEnumeration.pdf slides])
# [[组合数学 (Fall 2017)/Generating functions|Generating functions | 生成函数]] ( [http://tcs.nju.edu.cn/slides/comb2017/GeneratingFunction.pdf slides])
# [[组合数学 (Fall 2017)/Generating functions|Generating functions | 生成函数]] ( [http://tcs.nju.edu.cn/slides/comb2017/GeneratingFunction.pdf slides])
# [[组合数学 (Fall 2017)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ( [http://tcs.nju.edu.cn/slides/comb2017/Polya.pdf slides])
# [[组合数学 (Fall 2017)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ( [http://tcs.nju.edu.cn/slides/comb2017/Polya.pdf slides]):  guest lecture by [http://cs.nju.edu.cn/zhangyuan/ Yuan Zhang]
# Sieve methods | 筛法
# [[组合数学 (Fall 2017)/Sieve methods|Sieve methods | 筛法]] ( [http://tcs.nju.edu.cn/slides/comb2017/PIE.pdf slides])
# Cayley's formula | Cayley公式
# [[组合数学 (Fall 2017)/Cayley's formula|Cayley's formula | Cayley公式]]( [http://tcs.nju.edu.cn/slides/comb2017/Cayley.pdf slides])
# Existence problems | 存在性问题
# [[组合数学 (Fall 2017)/Existence problems|Existence problems | 存在性问题]] ( [http://tcs.nju.edu.cn/slides/comb2017/Existence.pdf slides])
# The probabilistic method | 概率法
# [[组合数学 (Fall 2017)/The probabilistic method|The probabilistic method | 概率法]]( [http://tcs.nju.edu.cn/slides/comb2017/ProbMethod.pdf slides])
# Extremal graph theory | 极值图论
# [[组合数学 (Fall 2017)/Extremal graph theory|Extremal graph theory | 极值图论]]( [http://tcs.nju.edu.cn/slides/comb2017/ExtremalGraphs.pdf slides])
# Extremal set theory | 极值集合论
# [[组合数学 (Fall 2017)/Extremal set theory|Extremal set theory | 极值集合论]]( [http://tcs.nju.edu.cn/slides/comb2017/ExtremalSets.pdf slides])
# Ramsey theory | Ramsey理论
# [[组合数学 (Fall 2017)/Ramsey theory|Ramsey theory | Ramsey理论]]( [http://tcs.nju.edu.cn/slides/comb2017/Ramsey.pdf slides])
# Matching theory | 匹配论
# [[组合数学 (Fall 2017)/Matching theory|Matching theory | 匹配论]]( [http://tcs.nju.edu.cn/slides/comb2017/Matchings.pdf slides])


= Concepts =
= Concepts =
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* [http://en.wikipedia.org/wiki/Generating_function Generating function] and [http://en.wikipedia.org/wiki/Formal_power_series formal power series]
* [http://en.wikipedia.org/wiki/Generating_function Generating function] and [http://en.wikipedia.org/wiki/Formal_power_series formal power series]
* [http://en.wikipedia.org/wiki/Binomial_series Newton's formula]
* [http://en.wikipedia.org/wiki/Binomial_series Newton's formula]
* [https://en.wikipedia.org/wiki/Burnside%27s_lemma Burnside's lemma]
**[https://en.wikipedia.org/wiki/Group_action group action] and [https://en.wikipedia.org/wiki/Group_action#Orbits_and_stabilizers orbits]
* [https://en.wikipedia.org/wiki/Permutation#Cycle_notation Cycle decomposition] of permutation
* [https://en.wikipedia.org/wiki/Pólya_enumeration_theorem Pólya enumeration theorem]
* [http://en.wikipedia.org/wiki/Inclusion-exclusion_principle The principle of inclusion-exclusion] (and more generally the [http://en.wikipedia.org/wiki/Sieve_theory sieve method])
* [http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula Möbius inversion formula]
* [http://en.wikipedia.org/wiki/Derangement Derangement], and [http://en.wikipedia.org/wiki/M%C3%A9nage_problem Problème des ménages]
* [http://en.wikipedia.org/wiki/Ryser%27s_formula#Ryser_formula Ryser's formula]
* [http://en.wikipedia.org/wiki/Euler_totient Euler totient function]
* [http://en.wikipedia.org/wiki/Cayley_formula Cayley's formula]
** [http://en.wikipedia.org/wiki/Prüfer_sequence Prüfer code for trees]
** [http://en.wikipedia.org/wiki/Kirchhoff%27s_matrix_tree_theorem Kirchhoff's matrix-tree theorem]
* [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/Pigeonhole_principle Pigeonhole principle]
:* [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/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/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]
* Two analytic inequalities:
:*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality]
:* 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)
* [https://en.wikipedia.org/wiki/Sunflower_(mathematics) Sunflower lemma and conjecture]
* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
* [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
* [https://en.wikipedia.org/wiki/Sauer–Shelah_lemma Sauer's lemma] and [https://en.wikipedia.org/wiki/VC_dimension VC dimension]
* [https://en.wikipedia.org/wiki/Kruskal–Katona_theorem Kruskal–Katona theorem]
* [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/Happy_Ending_problem Happy Ending problem]
* [https://en.wikipedia.org/wiki/Hall%27s_marriage_theorem Hall's theorem ] (the marriage theorem)
:* [https://en.wikipedia.org/wiki/Doubly_stochastic_matrix 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/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
* The  [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
:* [https://en.wikipedia.org/wiki/Menger%27s_theorem Menger's theorem]
:* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]

Latest revision as of 07:38, 2 January 2018

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 Monday, 10am
仙II-504
Office hours Monday 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 webpage for the Combinatorics class of fall 2017. Students who take this class should check this page periodically for content updates and new announcements.

Announcement

  • (2017/9/4) 新学期第一次上课。
  • (2017/9/18) 第一次作业发布。10月9日课上交。
  • (2017/10/16) 第一次作业发布。10月30日课上交。
  • (2017/12/27) 考前习题讲解与作业答疑。1月6日下午2点计算机系224。
  • (2017/12/27) 期末考试定于1月7日下午2点整准时开始,地点在仙2-504。

Course info

  • Instructor : 尹一通
  • email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
  • office: 804
  • Class meeting: Monday 10am, 仙II-504.
  • Office hour: Monday 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 | 基本计数slides
  2. Generating functions | 生成函数slides
  3. Pólya's theory of counting | Pólya计数法slides): guest lecture by Yuan Zhang
  4. Sieve methods | 筛法slides
  5. Cayley's formula | Cayley公式slides
  6. Existence problems | 存在性问题slides
  7. The probabilistic method | 概率法slides
  8. Extremal graph theory | 极值图论slides
  9. Extremal set theory | 极值集合论slides
  10. Ramsey theory | Ramsey理论slides
  11. Matching theory | 匹配论slides

Concepts