課程信息
4.6
55 個評分
22 個審閱
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100% 在線

立即開始,按照自己的計劃學習。
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中級

中級

完成時間(小時)

完成時間大約為35 小時

建議:8 weeks, 10-12 hours per week...
可選語言

英語(English)

字幕:英語(English)...
100% 在線

100% 在線

立即開始,按照自己的計劃學習。
可靈活調整截止日期

可靈活調整截止日期

根據您的日程表重置截止日期。
中級

中級

完成時間(小時)

完成時間大約為35 小時

建議:8 weeks, 10-12 hours per week...
可選語言

英語(English)

字幕:英語(English)...

教學大綱 - 您將從這門課程中學到什麼

1
完成時間(小時)
完成時間為 11 小時

Introduction

...
Reading
1 個視頻(共 8 分鐘), 4 個閱讀材料
Video1 個視頻
Reading4 個閱讀材料
Course Overview10分鐘
Grading and Logistics10分鐘
Suggested Readings分鐘
About the Instructor10分鐘
完成時間(小時)
完成時間為 11 小時

Permutations and binomial coefficients

In this introductory lecture we discuss fundamental combinatorial constructions: we will see how to compute the number of words of fixed length in a given alphabet, the number of permutations of a finite set and the number of subsets with a given number of elements in a finite set. The latter numbers are called binomial coefficients; we will see how they appear in various combinatorial problems in this and forthcoming lectures. As an application of combinatorial methods, we also give a combinatorial proof of Fermat's little theorem....
Reading
7 個視頻(共 78 分鐘), 1 個測驗
Video7 個視頻
Words9分鐘
Permutations10分鐘
k-permutations8分鐘
Merry-go-rounds and Fermat’s little theorem 18分鐘
Merry-go-rounds and Fermat’s little theorem 211分鐘
Binomial coefficients14分鐘
The Pascal triangle16分鐘
Quiz1 個練習
Quiz 2分鐘
2
完成時間(小時)
完成時間為 11 小時

Binomial coefficients, continued. Inclusion and exclusion formula.

In the first part of this lecture we will see more applications of binomial coefficients, in particular, their appearance in counting multisets. The second part is devoted to the principle of inclusion and exclusion: a technique which allows us to find the number of elements in the union of several sets, given the cardinalities of all of their intersections. We discuss its applications to various combinatorial problem, including the computation of the number of permutations without fixed points (the derangement problem)....
Reading
7 個視頻(共 87 分鐘), 1 個測驗
Video7 個視頻
Balls in boxes and multisets 110分鐘
Balls in boxes and multisets 26分鐘
Integer compositions11分鐘
Principle of inclusion and exclusion: two examples12分鐘
Principle of inclusion and exclusion: general statement9分鐘
The derangement problem19分鐘
Quiz1 個練習
Quiz 3分鐘
3
完成時間(小時)
完成時間為 14 小時

Linear recurrences. The Fibonacci sequence

We start with a well-known "rabbit problem", which dates back to Fibonacci. Using the Fibonacci sequence as our main example, we discuss a general method of solving linear recurrences with constant coefficients....
Reading
11 個視頻(共 105 分鐘), 1 個閱讀材料, 1 個測驗
Video11 個視頻
Fibonacci numbers and the Pascal triangle7分鐘
Domino tilings8分鐘
Vending machine problem10分鐘
Linear recurrence relations: definition7分鐘
The characteristic equation8分鐘
Linear recurrence relations of order 211分鐘
The Binet formula11分鐘
Sidebar: the golden ratio9分鐘
Linear recurrence relations of arbitrary order8分鐘
The case of roots with multiplicities12分鐘
Reading1 個閱讀材料
Spoilers! Solutions for quizzes 2, 3, and 4.分鐘
Quiz1 個練習
Quiz 4分鐘
4
完成時間(小時)
完成時間為 13 小時

A nonlinear recurrence: many faces of Catalan numbers

In this lecture we introduce Catalan numbers and discuss several ways to define them: via triangulations of a polygon, Dyck paths and binary trees. We also prove an explicit formula for Catalan numbers....
Reading
7 個視頻(共 73 分鐘), 1 個閱讀材料, 1 個測驗
Video7 個視頻
Recurrence relation for triangulations11分鐘
The cashier problem9分鐘
Dyck paths5分鐘
Recurrence relations for Dyck paths9分鐘
Reflection trick and a formula for Catalan numbers12分鐘
Binary trees15分鐘
Reading1 個閱讀材料
Solutions10分鐘
4.6

熱門審閱

創建者 RAMar 30th 2018

Excellent selection of material and presentation; TAs were of great help as well. The techniques taught in this course will be a nice addition to my algorithms analysis toolbox.

創建者 RRAug 22nd 2017

Great lectures and content. I really enjoyed it. However, the solutions exercises could be clearer and in more detail. Thank you!

講師

Avatar

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

關於 National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

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