User contributions for Etone

11:3611:36, 13 April 2023 diff hist +17,231‎ N 组合数学 (Fall 2023)/Cayley's formula ‎ Created page with "== Cayley's Formula == We now present a theorem of the number of labeled trees on a fixed number of vertices. It is due to [http://en.wikipedia.org/wiki/Arthur_Cayley Cayley] in 1889. The theorem is often referred by the name [http://en.wikipedia.org/wiki/Cayley's_formula Cayley's formula]. {{Theorem|Cayley's formula for trees| : There are <math>n^{n-2}</math> different trees on <math>n</math> distinct vertices. }} The theorem has several proofs, including the bijectio..." current
11:3611:36, 13 April 2023 diff hist +80‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
03:2603:26, 13 April 2023 diff hist 0‎ 概率论与数理统计 (Spring 2023)/Average-case analysis of QuickSort ‎ →‎快速排序算法的平均复杂度分析 II（基于期望的线性） current

14:2514:25, 29 March 2023 diff hist +19,696‎ N 组合数学 (Fall 2023)/Pólya's theory of counting ‎ Created page with "== Groups == A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisfies the following axioms: * ''closure'': <math>\forall g,h\in G, g\cdot h \in G</math>; * ''associativity'': <math>\forall f,g,h\in G, f\cdot(g\cdot h)=(f\cdot g)\cdot h</math>; * ''identity'': there exists a special element <math>e\in G</math>, called the '''identity''', such that <math>e\cdot g=g</math> for any <math>g\in G</math>; * ''inverse''..." current
14:2514:25, 29 March 2023 diff hist +105‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes
14:2414:24, 29 March 2023 diff hist −33‎ 概率论与数理统计 (Spring 2023)/Problem Set 2 ‎No edit summary

12:4612:46, 16 March 2023 diff hist +772‎ 组合数学 (Spring 2023) ‎ →‎Concepts
12:4512:45, 16 March 2023 diff hist +33,970‎ N 组合数学 (Fall 2023)/Sieve methods ‎ Created page with "== Principle of Inclusion-Exclusion == Let <math>A</math> and <math>B</math> be two finite sets. The cardinality of their union is :<math>|A\cup B|=|A|+|B|-{\color{Blue}|A\cap B|}</math>. For three sets <math>A</math>, <math>B</math>, and <math>C</math>, the cardinality of the union of these three sets is computed as :<math>|A\cup B\cup C|=|A|+|B|+|C|-{\color{Blue}|A\cap B|}-{\color{Blue}|A\cap C|}-{\color{Blue}|B\cap C|}+{\color{Red}|A\cap B\cap C|}</math>. This is illu..." current
12:4512:45, 16 March 2023 diff hist +68‎ 组合数学 (Spring 2023) ‎ →‎Lecture Notes

08:3108:31, 3 March 2023 diff hist +24,916‎ N 组合数学 (Fall 2023)/Generating functions ‎ Created page with "== Generating Functions == In Stanley's magnificent book ''Enumerative Combinatorics'', he comments the generating function as "the most useful but most difficult to understand method (for counting)". The solution to a counting problem is usually represented as some <math>a_n</math> depending a parameter <math>n</math>. Sometimes this <math>a_n</math> is called a ''counting function'' as it is a function of the parameter <math>n</math>. <math>a_n</math> can also be tre..."

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