组合数学 (Spring 2023): Difference between revisions

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|header6 =
|header6 =
|label6  = Class meetings
|label6  = Class meetings
|data6  = Friday, 2pm-4pm <br> 仙II-110
|data6  = Friday, 10am-12pm <br> 仙II-110
|header7 =
|header7 =
|label7  = Place
|label7  = Place
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= Announcement =
= Announcement =
* TBA
* '''(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 97: Line 100:


= Assignments =
= Assignments =
* TBA
* [[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 =
# TBA
# [[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 =
Line 109: Line 127:


= 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

组合数学
Combinatorics
Instructor
尹一通
Email 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.
v · d · e

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

  • email: yinyt@nju.edu.cn
  • office: 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

Lecture Notes

  1. Basic enumeration | 基本计数 (slides)
  2. Generating functions | 生成函数 (slides)
  3. Sieve methods | 筛法 (slides)
  4. Pólya's theory of counting | Pólya计数法 (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）

Resources

Concepts

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