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| {{Infobox
| | '''William Donald Hamilton''' [[Royal Society|FRS]] (1 August 1936 – 7 March 2000) was an [[English people|English]] [[evolutionary biology|evolutionary biologist]] whom [[Richard Dawkins]] praised as one of the greatest [[evolution]]ary theorists of the 20th century.<ref>[http://www.edge.org/3rd_culture/hamilton/hamilton_index.html Obituary by Richard Dawkins – ''The Independent'' – 10 March 2000]</ref> |
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| |title = <font size=3>组合数学 <br> | |
| Combinatorics</font>
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| |image = [[File:LW-combinatorics.jpeg|border|100px]] | | Hamilton became famous through his [[Theory|theoretical]] work on [[kin selection]] and [[altruism]]. He explained its [[Genetics|genetic]] basis, and this was a key part of the gene-centered view of [[evolution]]. In doing this, he became one of the forerunners of [[sociobiology]], as popularized by [[E.O. Wilson]]. Hamilton was certainly a big influence on Dawkins. He also published important work on [[sex ratio]]s and the [[evolution of sex]]. From 1984 to his death in 2000, he was the [[Royal Society]] Research Professor at [[Oxford University]]. He died of [[malaria]] contracted in the [[Democratic Republic of the Congo]]. |
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| |header1 =Instructor
| | == Hamilton's equation == |
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| | Hamilton's equation describes whether or not a gene for altruistic behaviour will spread in a population.<ref>Hamilton W.D. 1996. ''Narrow roads of geneland: the collected papers of W.D. Hamilton'', vol 1. Freeman, Oxford.</ref> The gene will spread if '''r'''x'''b''' is greater than '''c''': |
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| | :<math>rb > c \ </math> |
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| | where: |
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| | * <math>c \ </math> is the reproductive cost to the altruist, |
| |data2 = 尹一通
| | * <math>b \ </math> is the reproductive benefit to the recipient of the altruistic behavior, and |
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| | * <math>r \ </math> is the probability, above the population average, of the individuals sharing an altruistic gene – the "degree of relatedness". |
| |label3 = Email
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| |data3 = yitong.yin@gmail.com yinyt@nju.edu.cn
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| |data4= 计算机系 804
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| |header5 = Class
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| |label6 = Class meetings
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| |data6 = Thursday, 10am-12pm <br> 仙逸B-104
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| |label7 = Place
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| |label8 = Office hours
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| |data8 = Wednesday, 2-5pm <br>计算机系 804
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| |header9 = Textbook
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| |data10 = ''van Lint and Wilson,'' <br> A course in Combinatorics, 2nd Ed, <br> Cambridge Univ Press, 2001.
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| | == Collected papers == |
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| | Hamilton started to publish his collected papers starting in 1996, with short essays giving each paper context. He died after the preparation of the second volume, so the commentaries for the third volume came from his coauthors. |
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| This is the page for the class ''Combinatorics'' for the Fall 2011 semester. Students who take this class should check this page periodically for content updates and new announcements.
| | * Hamilton W.D. 1996. ''Narrow roads of gene land vol. 1: Evolution of social behaviour''. Freeman, Oxford. ISBN 0-7167-4530-5 |
| | * Hamilton W.D. 2002. ''Narrow roads of gene land vol. 2: Evolution of sex''. Oxford University Press, Oxford. ISBN 0-19-850336-9 |
| | * Hamilton W.D. 2005. ''Narrow roads of gene land, vol. 3: Last words'' (with essays by coauthors, ed. M. Ridley). Oxford University Press, Oxford. ISBN 0-19-856690-5 |
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| = Announcement = | | == References == |
| * (10/28/2011) <font size=5 color=red>第三次作业的第二题做了点修改，请同学们注意。</font>
| | {{Reflist}} |
| * (10/27/2011) <font size=3 color=red>第三次作业发布。两周后11月3日上课时交。</font>
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| * (09/29/2011) 第二次作业发布。两周后10月13日上课时交。
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| * (09/15/2011) 第一次作业发布，在Assignments部分。下周四上课时交。
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| * 由于有事需要外出，9月14日星期三下午的office hour改在9月13日下午。
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| * 第一、二次课的slides已发布，见lecture notes部分。
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| = Course info =
| | {{DEFAULTSORT:Hamilton, William Donald}} |
| * '''Instructor ''': 尹一通
| | [[Category:1936 births]] |
| :*email: yitong.yin@gmail.com, yinyt@nju.edu.cn,
| | [[Category:2000 deaths]] |
| :*office: 804
| | [[Category:English mathematicians]] |
| * '''Teaching fellow''': TBA
| | [[Category:Geneticists]] |
| :*email: TBA
| | [[Category:English evolutionary biologists]] |
| * '''Class meeting''': Thursday 10am-12pm, 仙逸B-104.
| | [[Category:Fellows of the Royal Society]] |
| * '''Office hour''': Wednesday 2-5pm, 计算机系 804.
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| = Syllabus =
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| === 先修课程 Prerequisites ===
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| * 离散数学（Discrete Mathematics）
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| * 线性代数（Linear Algebra）
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| * 概率论（Probability Theory）
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| === Course materials ===
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| * [[组合数学 (Fall 2011)/Course materials|教材和参考书清单]]
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| === 成绩 Grades ===
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| * 课程成绩：本课程将会有六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。
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| * 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。
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| === <font color=red> 学术诚信 Academic Integrity </font>===
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| 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。
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| 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”。
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| 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。
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| 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。
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| = Assignments =
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| * (2011/09/15) [[组合数学 (Fall 2011)/Problem set 1|Problem set 1]] due on Sept 22, in class.
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| * (2011/09/29) [[组合数学 (Fall 2011)/Problem set 2|Problem set 2]] due on Oct 13, in class.
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| * (2011/10/27) [[组合数学 (Fall 2011)/Problem set 3|Problem set 3]] due on Nov 3, in class.
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| = Lecture Notes =
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| # [[组合数学 (Fall 2011)/Basic enumeration|Basic enumeration]] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb1.pdf slides1] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb2-1.pdf slides2]
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| # [[组合数学 (Fall 2011)/Generating functions|Generating functions]] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb2-2.pdf slides1] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb3.pdf slides2]
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| # [[组合数学 (Fall 2011)/Sieve methods|Sieve methods]] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb4.pdf slides1]
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| # [[组合数学 (Fall 2011)/Pólya's theory of counting|Pólya's theory of counting]]
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| # [[组合数学 (Fall 2011)/Counting and existence|Counting and existence]] | [ftp://tcs.nju.edu.cn/slides/comb2011/comb6.pdf slides1]
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| # [[组合数学 (Fall 2011)/Discrete probability|Discrete probability]]
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| # [[组合数学 (Fall 2011)/The probabilistic method|The probabilistic method]]
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| # [[组合数学 (Fall 2011)/Extremal graph theory| Extremal graph theory]]
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| # [[组合数学 (Fall 2011)/Extremal set theory|Extremal set theory]]
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| # [[组合数学 (Fall 2011)/Ramsey theory|Ramsey theory]]
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| # [[组合数学 (Fall 2011)/Matching theory|Matching theory]]
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| # [[组合数学 (Fall 2011)/Flow and matching | Flow and matching]]
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| # [[组合数学 (Fall 2011)/Optimization|Optimization]]
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| # [[组合数学 (Fall 2011)/Matroid|Matroid]]
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| = Concepts =
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| * [http://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]
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| * [http://en.wikipedia.org/wiki/Composition_(number_theory) Composition of a number]
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| * [http://en.wikipedia.org/wiki/Combination#Number_of_combinations_with_repetition Combinations with repetition], [http://en.wikipedia.org/wiki/Multiset_coefficient#Multiset_coefficients <math>k</math>-multisets on a set]
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| * [http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind Stirling number of the second kind]
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| * [http://en.wikipedia.org/wiki/Partition_(number_theory) Partition of a number]
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| * [http://en.wikipedia.org/wiki/Partition_(number_theory)#Ferrers_diagram Ferrers diagram] (and the MathWorld [http://mathworld.wolfram.com/FerrersDiagram.html link])
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| * [http://en.wikipedia.org/wiki/Twelvefold_way The twelvefold way]
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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]
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| * [http://en.wikipedia.org/wiki/Fibonacci_number Fibonacci number]
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| * [http://en.wikipedia.org/wiki/Binomial_series Newton's formula]
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| * [http://en.wikipedia.org/wiki/Catalan_number Catalan number]
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| * [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])
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| * [http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula Möbius inversion formula]
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| * [http://en.wikipedia.org/wiki/Derangement Derangement], and [http://en.wikipedia.org/wiki/M%C3%A9nage_problem Problème des ménages]
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| * [http://en.wikipedia.org/wiki/Burnside's_lemma Burnside's lemma]
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| * [http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem Pólya enumeration theorem]
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| * [http://en.wikipedia.org/wiki/Double_counting_(proof_technique) Double counting] and the [http://en.wikipedia.org/wiki/Handshaking_lemma handshaking lemma]
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| * [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]
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| * [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]
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| * [http://en.wikipedia.org/wiki/Pigeonhole_principle Pigeonhole principle]
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| :* [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Szekeres_theorem Erdős–Szekeres theorem]
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| :* [http://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem Dirichlet's approximation theorem]
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| * [http://en.wikipedia.org/wiki/Probabilistic_method The Probabilistic Method]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model Erdős–Rényi model for random graphs]
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| * [http://en.wikipedia.org/wiki/Graph_property Graph property]
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| * Some graph parameters: [http://en.wikipedia.org/wiki/Girth_(graph_theory) girth <math>g(G)</math>], [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number <math>\chi(G)</math>], [http://mathworld.wolfram.com/IndependenceNumber.html Independence number <math>\alpha(G)</math>], [http://mathworld.wolfram.com/CliqueNumber.html clique number <math>\omega(G)</math>]
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| * [http://en.wikipedia.org/wiki/Extremal_graph_theory Extremal graph theory]
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| * [http://en.wikipedia.org/wiki/Turan_theorem Turán's theorem], [http://en.wikipedia.org/wiki/Tur%C3%A1n_graph Turán graph]
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| * Two analytic inequalities:
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| :*[http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality Cauchy–Schwarz inequality]
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| :* the [http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means inequality of arithmetic and geometric means]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem Erdős–Stone theorem] (fundamental theorem of extremal graph theory)
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| * [http://en.wikipedia.org/wiki/Dirac's_theorem Dirac's theorem]
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| * [http://en.wikipedia.org/wiki/Hall's_theorem Hall's theorem ] (the marriage theorem)
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| * [http://en.wikipedia.org/wiki/Birkhoff-Von_Neumann_theorem Birkhoff-Von Neumann theorem]
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| * [http://en.wikipedia.org/wiki/K%C3%B6nig's_theorem_(graph_theory) König-Egerváry theorem]
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| * [http://en.wikipedia.org/wiki/Dilworth's_theorem Dilworth's theorem]
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| * [http://en.wikipedia.org/wiki/Sperner_family Sperner system]
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| * [http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado_theorem Erdős–Ko–Rado theorem]
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| * [http://en.wikipedia.org/wiki/VC_dimension VC dimension]
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| * [http://en.wikipedia.org/wiki/Kruskal%E2%80%93Katona_theorem Kruskal–Katona theorem]
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| * [http://en.wikipedia.org/wiki/Ramsey_theory Ramsey theory]
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| :*[http://en.wikipedia.org/wiki/Ramsey's_theorem Ramsey's theorem]
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| :*[http://en.wikipedia.org/wiki/Happy_Ending_problem Happy Ending problem]
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| :*[http://en.wikipedia.org/wiki/Van_der_Waerden's_theorem Van der Waerden's theorem]
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| :*[http://en.wikipedia.org/wiki/Hales-Jewett_theorem Hales–Jewett theorem]
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| * [http://en.wikipedia.org/wiki/Lov%C3%A1sz_local_lemma Lovász local lemma]
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| * [http://en.wikipedia.org/wiki/Combinatorial_optimization Combinatorial optimization]
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| :* [http://en.wikipedia.org/wiki/Optimization_(mathematics) optimization]
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| :* [http://en.wikipedia.org/wiki/Convex_combination convex combination], [http://en.wikipedia.org/wiki/Convex_set convex set], [http://en.wikipedia.org/wiki/Convex_function convex function]
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| :* [http://en.wikipedia.org/wiki/Local_optimum local optimum] (see also [http://en.wikipedia.org/wiki/Maxima_and_minima maxima and minima])
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| * [http://en.wikipedia.org/wiki/Linear_programming Linear programming]
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| :* [http://en.wikipedia.org/wiki/Linear_inequality linear constraint]
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| :* [http://en.wikipedia.org/wiki/Hyperplane hyperplane], [http://en.wikipedia.org/wiki/Half_space halfspace], [http://en.wikipedia.org/wiki/Polyhedron polyhedron], [http://en.wikipedia.org/wiki/Convex_polytope convex polytope]
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| :* [http://en.wikipedia.org/wiki/Simplex_algorithm the Simplex algorithm]
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| * The [http://en.wikipedia.org/wiki/Max-flow_min-cut_theorem Max-Flow Min-Cut Theorem]
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| :* [http://en.wikipedia.org/wiki/Maximum_flow_problem Maximum flow]
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| :* [http://en.wikipedia.org/wiki/Minimum_cut minimum cut]
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| * [http://en.wikipedia.org/wiki/Unimodular_matrix Unimodularity]
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| * [http://en.wikipedia.org/wiki/Dual_linear_program Duality]
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| :* [http://en.wikipedia.org/wiki/Linear_programming#Duality LP Duality]
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| * [http://en.wikipedia.org/wiki/Matroid Matroid]
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| :* [http://en.wikipedia.org/wiki/Weighted_matroid weighted matroid] and [http://en.wikipedia.org/wiki/Greedy_algorithm greedy algorithm]
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| :* [http://en.wikipedia.org/wiki/Matroid_intersection Matroid intersection]
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| * [http://en.wikipedia.org/wiki/Laplacian_matrix Laplacian]
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| * [http://en.wikipedia.org/wiki/Algebraic_connectivity <math>\lambda_2</math> of a graph] and [http://en.wikipedia.org/wiki/Expander_graph#Cheeger_Inequalities Cheeger Inequalities]
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| * [http://en.wikipedia.org/wiki/Expander_graph Expander graph]
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| * [http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma Szemerédi regularity lemma]
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