组合数学 (Spring 2023): Difference between revisions
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|data3 = | |data3 = yinyt@nju.edu.cn | ||
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|label4= office | |label4= office | ||
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|label6 = Class meetings | |label6 = Class meetings | ||
|data6 = Friday, | |data6 = Friday, 10am-12pm <br> 仙II-110 | ||
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|label7 = Place | |label7 = Place | ||
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= Announcement = | = Announcement = | ||
* | * '''(2023/03/16)'''<font color=red size=4> 第一次作业已发布</font>，请在 2023/03/24 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A1.pdf'). | ||
* '''(2023/04/13)'''<font color=red size=4> 第二次作业已发布</font>，请在 2023/04/21 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A2.pdf'). | |||
* '''(2023/05/09)'''<font color=red size=4> 第三次作业已发布</font>，请在 2023/05/19 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A3.pdf'). | |||
* '''(2023/06/08)'''<font color=red size=4> 第四次作业已发布</font>，请在 2023/06/16 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A4.pdf'). | |||
= Course info = | = Course info = | ||
| Line 66: | Line 69: | ||
:*'''email''': yinyt@nju.edu.cn | :*'''email''': yinyt@nju.edu.cn | ||
:*'''office''': 计算机系 804 | :*'''office''': 计算机系 804 | ||
* '''Teaching assistant''': | * '''Teaching assistant''': | ||
** [https://lhy-gispzjz.github.io 刘弘洋] ([mailto: | ** [https://lhy-gispzjz.github.io 刘弘洋] ([mailto:liuhongyang@smail.nju.edu.cn liuhongyang@smail.nju.edu.cn]) | ||
** [https://wcysai.com 王淳扬] ([mailto:wcysai@smail.nju.edu.cn wcysai@smail.nju.edu.cn]) | ** [https://wcysai.com 王淳扬] ([mailto:wcysai@smail.nju.edu.cn wcysai@smail.nju.edu.cn]) | ||
* '''Class meeting''': Friday, | * '''Class meeting''': Friday, 10am-12pm, 仙II-110. | ||
* '''Office hour''': Thursday, 2pm-4pm, 计算机系 804. | * '''Office hour''': Thursday, 2pm-4pm, 计算机系 804. | ||
:* '''QQ群''': 542456114 (加入时需报姓名、专业、学号) | :* '''QQ群''': 542456114 (加入时需报姓名、专业、学号) | ||
| Line 97: | Line 100: | ||
= Assignments = | = Assignments = | ||
* | * [[组合数学 (Fall 2023)/Problem Set 1|Problem Set 1]]，请在 2023/03/24 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A1.pdf'). [[组合数学 (Spring 2023)/第一次作业提交名单|第一次作业提交名单]] | ||
* [[组合数学 (Fall 2023)/Problem Set 2|Problem Set 2]]，请在 2023/04/21 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A2.pdf'). [[组合数学 (Spring 2023)/第二次作业提交名单|第二次作业提交名单]] | |||
* [[组合数学 (Fall 2023)/Problem Set 3|Problem Set 3]]，请在 2023/05/19 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A3.pdf'). [[组合数学 (Spring 2023)/第三次作业提交名单|第三次作业提交名单]] | |||
* [[组合数学 (Fall 2023)/Problem Set 4|Problem Set 4]]，请在 2023/06/16 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名为'学号_姓名_A4.pdf'). [[组合数学 (Spring 2023)/第四次作业提交名单|第四次作业提交名单]] | |||
= Lecture Notes = | = Lecture Notes = | ||
# | # [[组合数学 (Fall 2023)/Basic enumeration|Basic enumeration | 基本计数]] ([http://tcs.nju.edu.cn/slides/comb2023/BasicEnumeration.pdf slides]) | ||
# [[组合数学 (Fall 2023)/Generating functions|Generating functions | 生成函数]] ([http://tcs.nju.edu.cn/slides/comb2023/GeneratingFunction.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Sieve methods|Sieve methods | 筛法]] ([http://tcs.nju.edu.cn/slides/comb2023/PIE.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ([http://tcs.nju.edu.cn/slides/comb2023/Polya.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Cayley's formula|Cayley's formula | Cayley公式]] ([http://tcs.nju.edu.cn/slides/comb2023/Cayley.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Existence problems|Existence problems | 存在性问题]] ([http://tcs.nju.edu.cn/slides/comb2023/Existence.pdf slides]) | |||
# [[组合数学 (Fall 2023)/The probabilistic method|The probabilistic method | 概率法]] ([http://tcs.nju.edu.cn/slides/comb2023/ProbMethod.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Extremal graph theory|Extremal graph theory | 极值图论]] ([http://tcs.nju.edu.cn/slides/comb2023/ExtremalGraphs.pdf slides]) | |||
# [[组合数学 (Fall 2023)/Extremal set theory|Extremal set theory | 极值集合论]]（ [http://tcs.nju.edu.cn/slides/comb2023/ExtremalSets.pdf slides]） | |||
#* [https://mathweb.ucsd.edu/~ronspubs/90_03_erdos_ko_rado.pdf Old and new proofs of the Erdős–Ko–Rado theorem] by Frankl and Graham | |||
#* [https://arxiv.org/pdf/1908.08483.pdf Improved bounds for the sunflower lemma] by Alweiss-Lovet-Wu-Zhang and a [https://arxiv.org/pdf/1909.04774.pdf simplified proof] by Rao | |||
# [[组合数学 (Fall 2023)/Ramsey theory|Ramsey theory | Ramsey理论]]（ [http://tcs.nju.edu.cn/slides/comb2023/Ramsey.pdf slides]） | |||
# [[组合数学 (Fall 2023)/Matching theory|Matching theory | 匹配论]]（ [http://tcs.nju.edu.cn/slides/comb2023/Matchings.pdf slides]） | |||
= Resources = | = Resources = | ||
* [http://math.mit.edu/~fox/MAT307.html Combinatorics course] by Jacob Fox | * [http://math.mit.edu/~fox/MAT307.html Combinatorics course] by Jacob Fox | ||
* [https://yufeizhao.com/pm/ Probabilistic Methods in Combinatorics] and [https://yufeizhao.com/gtacbook/ Graph Theory and Additive Combinatorics] by Yufei Zhao | |||
* [https://www.math.uvic.ca/~noelj/combinatoricsLectures.html Combinatorics Lecture Videos online] | |||
* [https://www.math.ucla.edu/~pak/lectures/Math-Videos/comb-videos.htm Collection of Combinatorics Videos] | * [https://www.math.ucla.edu/~pak/lectures/Math-Videos/comb-videos.htm Collection of Combinatorics Videos] | ||
= Concepts = | = Concepts = | ||
* [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient] | |||
* [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] | |||
* [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] | |||
* [https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem] | |||
* [https://en.wikipedia.org/wiki/Sperner%27s_theorem Sperner's theorem] | |||
** [https://en.wikipedia.org/wiki/Sperner_family Sperner system] or '''antichain''' | |||
* [https://en.wikipedia.org/wiki/Sauer%E2%80%93Shelah_lemma Sauer–Shelah lemma] | |||
** [https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_dimension Vapnik–Chervonenkis dimension] | |||
* [https://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's_theorem Ramsey's theorem] | |||
:*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem] | |||
:*[https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem Van der Waerden's theorem] | |||
:*[https://en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem Hales–Jewett theorem] | |||
* [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] | |||
* [https://en.wikipedia.org/wiki/Linear_programming Linear programming] | |||
** [https://en.wikipedia.org/wiki/Dual_linear_program Duality] | |||
** [https://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity] | |||
* [https://en.wikipedia.org/wiki/Matroid Matroid] | |||
Latest revision as of 18:58, 9 April 2024
| Instructor | |
|---|---|
| 尹一通 | |
| yinyt@nju.edu.cn | |
| office | 计算机系 804 |
| Class | |
| Class meetings |
Friday, 10am-12pm 仙II-110 |
| Office hours |
Thursday, 2pm-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. | |
This is the webpage for the Combinatorics class of Spring 2023. Students who take this class should check this page periodically for content updates and new announcements.
Announcement
- (2023/03/16) 第一次作业已发布，请在 2023/03/24 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A1.pdf').
- (2023/04/13) 第二次作业已发布，请在 2023/04/21 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A2.pdf').
- (2023/05/09) 第三次作业已发布，请在 2023/05/19 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A3.pdf').
- (2023/06/08) 第四次作业已发布，请在 2023/06/16 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A4.pdf').
Course info
- Instructor : 尹一通 (homepage)
- email: yinyt@nju.edu.cn
- office: 计算机系 804
- Teaching assistant:
- Class meeting: Friday, 10am-12pm, 仙II-110.
- Office hour: Thursday, 2pm-4pm, 计算机系 804.
- QQ群: 542456114 (加入时需报姓名、专业、学号)
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，请在 2023/03/24 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A1.pdf'). 第一次作业提交名单
- Problem Set 2，请在 2023/04/21 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A2.pdf'). 第二次作业提交名单
- Problem Set 3，请在 2023/05/19 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A3.pdf'). 第三次作业提交名单
- Problem Set 4，请在 2023/06/16 上课之前提交到 njucomb23@163.com (文件名为'学号_姓名_A4.pdf'). 第四次作业提交名单
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 (slides)
- Cayley's formula | Cayley公式 (slides)
- Existence problems | 存在性问题 (slides)
- The probabilistic method | 概率法 (slides)
- Extremal graph theory | 极值图论 (slides)
- Extremal set theory | 极值集合论（ slides）
- Old and new proofs of the Erdős–Ko–Rado theorem by Frankl and Graham
- Improved bounds for the sunflower lemma by Alweiss-Lovet-Wu-Zhang and a simplified proof by Rao
- Ramsey theory | Ramsey理论（ slides）
- Matching theory | 匹配论（ slides）
Resources
- Combinatorics course by Jacob Fox
- Probabilistic Methods in Combinatorics and Graph Theory and Additive Combinatorics by Yufei Zhao
- Combinatorics Lecture Videos online
- 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
- 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's theorem
- Sperner system or antichain
- Sauer–Shelah lemma
- Kruskal–Katona theorem
- Ramsey theory
- Hall's theorem (the marriage theorem)