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14 May 2024
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N 17:15 | 组合数学 (Spring 2024)/Problem Set 3 3 changes history +2,925 [Roundgod; Gispzjz (2×)] | |||
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17:15 (cur | prev) −1 Roundgod talk contribs (→Problem 1) | ||||
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15:55 (cur | prev) +1 Gispzjz talk contribs (→Problem 5) | ||||
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15:55 (cur | prev) +2,925 Gispzjz talk contribs (Created page with "== Problem 1 == Solve the following two existence problems: * You are given <math>n</math> integers <math>a_1,a_2,\dots,a_n</math>, such that for each <math> 1\leq i\leq n</math> it holds that <math>i-n\leq a_i\leq i-1</math>. Show that there exists a '''nonempty''' subsequence (not necessarily consecutive) of these integers, whose sum is equal to <math> 0 </math>. (Hint: Consider <math> b_i=a_i-i </math>) * You are given two '''multisets''' <math> A </math> and <ma...") | |||
| N 13:06 | 组合数学 (Spring 2024)/Problem Set 2 diffhist +5,164 Roundgod talk contribs (Created page with "== Problem 1 == Count the number of lattice paths from <math>(0, 0)</math> to <math>(n, n)</math> with steps up and to the right, when two paths are considered equal if one can be moved on top of the other by a rotation or reflection. For example: center|frameless (Hint: Represent a path as a sequence of <math>0</math>s and <math>1</math>s and determine the group that acts on these sequences.) == Problem 2 == Suppose we are given <math>m</math> <math>n<...") | ||||
| N 13:05 | 组合数学 (Spring 2024)/Problem Set 1 diffhist +1,789 Roundgod talk contribs (Created page with "== Problem 1 == How many <math>n\times m</math> matrices of <math>0</math>'s and <math>1</math>'s are there, such that every row and column contains an even number of <math>1</math>'s? An odd number of <math>1</math>'s? == Problem 2 == * There is a set of <math>2n</math> people: <math>n</math> male and <math>n</math> female. A good party is a set with the same number of male and female. How many possibilities are there to build such a good party? * Try to express the a...") | ||||
| N 09:38 | 组合数学 (Fall 2024)/Extremal graph theory diffhist +18,939 Etone talk contribs (Created page with "== Forbidden Cliques == Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can have, if <math>G</math> has some property?" === Mantel's theorem === We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</math>. {{Theorem|Theorem (Mantel 1907)| :Suppose <math>G(V,E)</math> is graph on <math>n</math> vertice without triangles. Then <m...") | ||||