高级算法 (Fall 2019) and 组合数学 (Fall 2019): Difference between pages
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|title = <font size=3> | |title = <font size=3>组合数学 <br> | ||
<br> | Combinatorics</font> | ||
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|label3 = Email | |label3 = Email | ||
|data3 = | |data3 = yitong.yin@gmail.com yinyt@nju.edu.cn | ||
|header4 = | |header4 = | ||
|label4= office | |label4= office | ||
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|header6 = | |header6 = | ||
|label6 = Class meetings | |label6 = Class meetings | ||
|data6 = Wednesday, | |data6 = Wednesday, 2pm-4pm <br> 仙I-319 | ||
|header7 = | |header7 = | ||
|label7 = Place | |label7 = Place | ||
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|label8 = Office hours | |label8 = Office hours | ||
|data8 = Wednesday, 4pm-6pm <br>804 | |data8 = Wednesday, 4pm-6pm <br>计算机系 804 | ||
|header9 = | |header9 = Textbook | ||
|label9 = | |label9 = | ||
|data9 = | |data9 = | ||
|header10 = | |header10 = | ||
|label10 = | |label10 = | ||
|data10 = [[File: | |data10 = [[File:LW-combinatorics.jpeg|border|100px]] | ||
|header11 = | |header11 = | ||
|label11 = | |label11 = | ||
|data11 = | |data11 = van Lint and Wilson. <br> ''A course in Combinatorics, 2nd ed.'', <br> Cambridge Univ Press, 2001. | ||
|header12 = | |header12 = | ||
|label12 = | |label12 = | ||
|data12 = [[File: | |data12 = [[File:Jukna_book.jpg|border|100px]] | ||
|header13 = | |header13 = | ||
|label13 = | |label13 = | ||
|data13 = | |data13 = Jukna. ''Extremal Combinatorics: <br> With Applications in Computer Science,<br>2nd ed.'', Springer, 2011. | ||
|belowstyle = background:#ddf; | |belowstyle = background:#ddf; | ||
|below = | |below = | ||
}} | }} | ||
This is the webpage for the '' | This is the webpage for the ''Combinatorics'' class of fall 2019. Students who take this class should check this page periodically for content updates and new announcements. | ||
= Announcement = | = Announcement = | ||
*<font size=5 color=red>由于学校网络中心没有开放外网对Mathoid端口的访问权,因此目前讲义只能在校内访问时正常显示数学公式。</font> | |||
* (2019/9/6) 第一课的lecture notes和slides已经发布。 | * (2019/9/6) 第一课的lecture notes和slides已经发布。 | ||
= Course info = | = Course info = | ||
* '''Instructor ''': 尹一通 ([http://tcs.nju.edu.cn/yinyt/ homepage]) | * '''Instructor ''': 尹一通 ([http://tcs.nju.edu.cn/yinyt/ homepage]) | ||
:* | :*email: yinyt@nju.edu.cn | ||
* '''Teaching | :*office: 804 | ||
:* | * '''Teaching assistant''': 陈海敏 ([http://tcs.nju.edu.cn/files/people/haimin/ homepage]) | ||
* '''Class meeting''': Wednesday | :*email: ichenhm@gmail.com | ||
* '''Office hour''': Wednesday 4pm-6pm, 计算机系 804. | * '''Class meeting''': Wednesday, 2pm-4pm, 仙I-319. | ||
* '''Office hour''': Wednesday, 4pm-6pm, 计算机系 804. | |||
= Syllabus = | = Syllabus = | ||
=== 先修课程 Prerequisites === | === 先修课程 Prerequisites === | ||
* | * 离散数学(Discrete Mathematics) | ||
* | * 线性代数(Linear Algebra) | ||
* 概率论(Probability Theory) | |||
=== Course materials === | === Course materials === | ||
* [[ | * [[组合数学 (Fall 2019)/Course materials|<font size=3>教材和参考书清单</font>]] | ||
=== 成绩 Grades === | === 成绩 Grades === | ||
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= Assignments = | = Assignments = | ||
*[[ | *[[组合数学 (Fall 2019)/Problem Set 1|Problem Set 1]] due on Sept 25, in class. | ||
= Lecture Notes = | = Lecture Notes = | ||
# [[ | # [[组合数学 (Fall 2019)/Basic enumeration|Basic enumeration | 基本计数]] ( [http://tcs.nju.edu.cn/slides/comb2019/BasicEnumeration.pdf slides]) | ||
# | # [[组合数学 (Fall 2019)/Generating functions|Generating functions | 生成函数]] ( [http://tcs.nju.edu.cn/slides/comb2019/GeneratingFunction.pdf slides]) | ||
# [[组合数学 (Fall 2019)/Sieve methods|Sieve methods | 筛法]] ( [http://tcs.nju.edu.cn/slides/comb2019/PIE.pdf slides]) | |||
# | # Pólya's theory of counting | Pólya计数法 | ||
# Cayley's formula | Cayley公式 | |||
# | # Existence problems | 存在性问题 | ||
# | # The probabilistic method | 概率法 | ||
# | # Extremal graph theory | 极值图论 | ||
# | # Extremal set theory | 极值集合论 | ||
# | # Ramsey theory | Ramsey理论 | ||
# | # Matching theory | 匹配论 | ||
# | |||
# | |||
= | = Resources = | ||
* [http:// | * [http://math.mit.edu/~fox/MAT307.html Combinatorics course] by Jacob Fox (now at Stanford) taught at MIT and Princeton. | ||
* [http:// | * [https://www.math.ucla.edu/~pak/lectures/Math-Videos/comb-videos.htm Collection of Combinatorics Videos] | ||
* [http:// | |||
* [https:// | = Concepts = | ||
* [http:// | * [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient] | ||
* The [https:// | * [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way] | ||
* [http://en.wikipedia.org/wiki/Composition_(number_theory) Composition of a number] | |||
* [http://en.wikipedia.org/wiki/Multiset#Formal_definition Multiset] | |||
* [http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition Combinations with repetition], [http://en.wikipedia.org/wiki/Multiset#Counting_multisets <math>k</math>-multisets on a set] | |||
* [http://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients Multinomial coefficients] | |||
* [http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling number of the second kind] | |||
* [http://en.wikipedia.org/wiki/Partition_(number_theory) Partition of a number] | |||
** [http://en.wikipedia.org/wiki/Young_tableau Young tableau] | |||
* [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number] | |||
* [http://en.wikipedia.org/wiki/Catalan_number Catalan number] | |||
* [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] | |||
* [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] |
Revision as of 06:15, 30 September 2019
This is the webpage for the Combinatorics class of fall 2019. Students who take this class should check this page periodically for content updates and new announcements.
Announcement
- 由于学校网络中心没有开放外网对Mathoid端口的访问权,因此目前讲义只能在校内访问时正常显示数学公式。
- (2019/9/6) 第一课的lecture notes和slides已经发布。
Course info
- Instructor : 尹一通 (homepage)
- email: yinyt@nju.edu.cn
- office: 804
- Teaching assistant: 陈海敏 (homepage)
- email: ichenhm@gmail.com
- Class meeting: Wednesday, 2pm-4pm, 仙I-319.
- Office hour: Wednesday, 4pm-6pm, 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 (≥ 60%) 和期末考试成绩 (≤ 40%) 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是你个人的贡献。在完成作业的过程中,允许讨论,前提是讨论的所有参与者均处于同等完成度。但关键想法的执行、以及作业文本的写作必须独立完成,并在作业中致谢(acknowledge)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
- Problem Set 1 due on Sept 25, in class.
Lecture Notes
- Basic enumeration | 基本计数 ( slides)
- Generating functions | 生成函数 ( slides)
- Sieve methods | 筛法 ( slides)
- Pólya's theory of counting | Pólya计数法
- Cayley's formula | Cayley公式
- Existence problems | 存在性问题
- The probabilistic method | 概率法
- Extremal graph theory | 极值图论
- Extremal set theory | 极值集合论
- Ramsey theory | Ramsey理论
- Matching theory | 匹配论
Resources
- Combinatorics course by Jacob Fox (now at Stanford) taught at MIT and Princeton.
- Collection of Combinatorics Videos
Concepts
- Binomial coefficient
- The twelvefold way
- Composition of a number
- Multiset
- Combinations with repetition, [math]\displaystyle{ k }[/math]-multisets on a set
- Multinomial coefficients
- Stirling number of the second kind
- Partition of a number
- Fibonacci number
- Catalan number
- Generating function and formal power series
- Newton's formula
- Burnside's lemma
- group action and orbits
- Cycle decomposition of permutation
- Pólya enumeration theorem
- The principle of inclusion-exclusion (and more generally the sieve method)
- Möbius inversion formula
- Derangement, and Problème des ménages
- Ryser's formula
- Euler totient function
- Cayley's formula
- Double counting and the handshaking lemma
- Sperner's lemma and Brouwer fixed point theorem
- Pigeonhole principle
- The Probabilistic Method
- Lovász local lemma
- Erdős–Rényi model for random graphs
- Extremal graph theory
- Turán's theorem, Turán graph
- Two analytic inequalities:
- Erdős–Stone theorem (fundamental theorem of extremal graph theory)
- Sunflower lemma and conjecture
- Erdős–Ko–Rado theorem
- Sperner system
- Sauer's lemma and VC dimension
- Kruskal–Katona theorem
- Ramsey theory
- Hall's theorem (the marriage theorem)