组合数学 (Spring 2025): Difference between revisions
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{{Infobox | |||
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|title = <font size=3>组合数学 <br> | |||
Combinatorics</font> | |||
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This is the webpage for the ''Combinatorics'' class of Spring | |image = | ||
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|header1 =Instructor | |||
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|data1 = | |||
|header2 = | |||
|label2 = | |||
|data2 = 尹一通 | |||
|header3 = | |||
|label3 = Email | |||
|data3 = yinyt@nju.edu.cn | |||
|header4 = | |||
|label4= office | |||
|data4= 计算机系 804 | |||
|header5 = Class | |||
|label5 = | |||
|data5 = | |||
|header6 = | |||
|label6 = Class meetings | |||
|data6 = Wednesday, 2pm-4pm <br> 逸A-322 | |||
|header7 = | |||
|label7 = Place | |||
|data7 = | |||
|header8 = | |||
|label8 = Office hours | |||
|data8 = TBA <br>计算机系 804 | |||
|header9 = Textbook | |||
|label9 = | |||
|data9 = | |||
|header10 = | |||
|label10 = | |||
|data10 = [[File:LW-combinatorics.jpeg|border|100px]] | |||
|header11 = | |||
|label11 = | |||
|data11 = van Lint and Wilson. <br> ''A course in Combinatorics, 2nd ed.'', <br> Cambridge Univ Press, 2001. | |||
|header12 = | |||
|label12 = | |||
|data12 = [[File:Jukna_book.jpg|border|100px]] | |||
|header13 = | |||
|label13 = | |||
|data13 = Jukna. ''Extremal Combinatorics: <br> With Applications in Computer Science,<br>2nd ed.'', Springer, 2011. | |||
|belowstyle = background:#ddf; | |||
|below = | |||
}} | |||
This is the webpage for the ''Combinatorics'' class of Spring 2025. Students who take this class should check this page periodically for content updates and new announcements. | |||
= Announcement = | = Announcement = | ||
* '''(2025/03/18)'''<font color=red size=4> 第一次作业已发布</font>，请在 2025/04/02 上课之前提交到 [mailto:njucomb25@163.com njucomb25@163.com] (文件名为'学号_姓名_A1.pdf') | |||
* '''(2025/04/09)'''<font color=red size=4> 第二次作业已发布</font>，请在 2025/04/23 上课之前提交到 [mailto:njucomb25@163.com njucomb25@163.com] (文件名为'学号_姓名_A2.pdf') | |||
= 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 | |||
:* '''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 | ** [https://lhy-gispzjz.github.io 刘弘洋] ([mailto:liuhongyang@smail.nju.edu.cn liuhongyang@smail.nju.edu.cn]) | ||
** 丁天行 | ** 丁天行 | ||
* '''Class meeting''': Wednesday, 2pm-4pm, 逸A-322. | * '''Class meeting''': Wednesday, 2pm-4pm, 逸A-322. | ||
* '''Office hour''': TBA | * '''Office hour''': TBA | ||
:* '''QQ群''': 260501949 (加入时需报姓名、专业、学号) | |||
:* '''QQ群''': | |||
= Syllabus = | = Syllabus = | ||
=== 先修课程 Prerequisites === | === 先修课程 Prerequisites === | ||
* 离散数学（Discrete Mathematics） | * 离散数学（Discrete Mathematics） | ||
* 线性代数（Linear Algebra） | * 线性代数（Linear Algebra） | ||
| Line 28: | Line 82: | ||
=== Course materials === | === Course materials === | ||
* [[组合数学 (Spring 2025)/Course materials|<font size=3>教材和参考书清单</font>]] | |||
* [ | |||
=== 成绩 Grades === | === 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。 | * 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。 | * 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。 | ||
=== <font color= | === <font color=red> 学术诚信 Academic Integrity </font>=== | ||
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。 | 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”。 | 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 [http://www.acm.org/publications/policies/plagiarism_policy 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，<font color= | 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 [http://www.acm.org/publications/policies/plagiarism_policy 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，<font color=red> 抄袭和被抄袭双方的成绩都将被取消</font>。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。 | 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 = | = Assignments = | ||
* [[组合数学 (Spring 2025)/Problem Set 1|Problem Set 1]] [[组合数学 (Spring 2025)/第一次作业提交名单|第一次作业提交名单]] | |||
* [[组合数学 (Spring 2025)/Problem Set 2|Problem Set 2]] | |||
= Lecture Notes = | = Lecture Notes = | ||
# [[组合数学 (Spring 2025)/Basic enumeration|Basic enumeration | 基本计数]] | # [[组合数学 (Spring 2025)/Basic enumeration|Basic enumeration | 基本计数]] ([http://tcs.nju.edu.cn/slides/comb2025/BasicEnumeration.pdf slides]) | ||
# [[组合数学 (Fall 2025)/Generating functions|Generating functions | 生成函数]] ([http://tcs.nju.edu.cn/slides/comb2025/GeneratingFunction.pdf slides]) | |||
# [[组合数学 (Fall 2025)/Sieve methods|Sieve methods | 筛法]] ([http://tcs.nju.edu.cn/slides/comb2025/PIE.pdf slides]) | |||
# [[组合数学 (Fall 2025)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ([http://tcs.nju.edu.cn/slides/comb2025/Polya.pdf slides]) | |||
# [[组合数学 (Fall 2025)/Cayley's formula|Cayley's formula | Cayley公式]] ([http://tcs.nju.edu.cn/slides/comb2025/Cayley.pdf slides]) | |||
# [[组合数学 (Fall 2025)/Existence problems|Existence problems | 存在性问题]] ([http://tcs.nju.edu.cn/slides/comb2025/Existence.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://yufeizhao.com/pm/ Probabilistic Methods in Combinatorics] and [https://yufeizhao.com/gtacbook/ Graph Theory and Additive Combinatorics] by Yufei Zhao | ||
| Line 58: | Line 116: | ||
= Concepts = | = Concepts = | ||
* [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient] | * [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient] | ||
* [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way] | * [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way] | ||
| Line 77: | Line 134: | ||
* [http://en.wikipedia.org/wiki/Ryser%27s_formula#Ryser_formula Ryser's formula] | * [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/Euler_totient Euler totient function] | ||
* [ | * [https://en.wikipedia.org/wiki/Burnside%27s_lemma Burnside's lemma] | ||
** [ | ** [https://en.wikipedia.org/wiki/Group_action Group action] | ||
** [ | ** [https://en.wikipedia.org/wiki/Group_action#Orbits_and_stabilizers Orbits] | ||
* [ | * [https://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem Pólya enumeration theorem] | ||
** [ | ** [https://en.wikipedia.org/wiki/Permutation_group Permutation group] | ||
** [ | ** [https://en.wikipedia.org/wiki/Cycle_index Cycle index] | ||
* [http://en.wikipedia.org/wiki/Cayley_formula Cayley's formula] | * [http://en.wikipedia.org/wiki/Cayley_formula Cayley's formula] | ||
** [http://en.wikipedia.org/wiki/ | ** [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/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/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/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/Lov%C3%A1sz_local_lemma Lovász local lemma] | ||
| Line 98: | Line 153: | ||
* [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] | ||
* Two analytic inequalities: | * Two analytic inequalities: | ||
:*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality] | |||
:* [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] | :* 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) | ||
* [ | * [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_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] | :*[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/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/Dilworth's_theorem Dilworth's theorem] | ||
:* [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] | ||
* The [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut 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] | :* [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 06:58, 9 April 2025
| Instructor | |
|---|---|
| 尹一通 | |
| yinyt@nju.edu.cn | |
| office | 计算机系 804 |
| Class | |
| Class meetings |
Wednesday, 2pm-4pm 逸A-322 |
| Office hours |
TBA 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 2025. Students who take this class should check this page periodically for content updates and new announcements.
Announcement
- (2025/03/18) 第一次作业已发布，请在 2025/04/02 上课之前提交到 njucomb25@163.com (文件名为'学号_姓名_A1.pdf')
- (2025/04/09) 第二次作业已发布，请在 2025/04/23 上课之前提交到 njucomb25@163.com (文件名为'学号_姓名_A2.pdf')
Course info
- Instructor : 尹一通 (homepage)
- email: yinyt@nju.edu.cn
- office: 计算机系 804
- Teaching assistant:
- Class meeting: Wednesday, 2pm-4pm, 逸A-322.
- Office hour: TBA
- QQ群: 260501949 (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
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)
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
- Burnside's lemma
- Pólya enumeration theorem
- 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)