课程信息
4.6
55 个评分
22 个审阅
100% online

100% online

立即开始,按照自己的计划学习。
可灵活调整截止日期

可灵活调整截止日期

根据您的日程表重置截止日期。
中级

中级

完成时间(小时)

完成时间大约为35 小时

建议:8 weeks, 10-12 hours per week...
可选语言

英语(English)

字幕:英语(English)...
100% online

100% online

立即开始,按照自己的计划学习。
可灵活调整截止日期

可灵活调整截止日期

根据您的日程表重置截止日期。
中级

中级

完成时间(小时)

完成时间大约为35 小时

建议:8 weeks, 10-12 hours per week...
可选语言

英语(English)

字幕:英语(English)...

教学大纲 - 您将从这门课程中学到什么

1
完成时间(小时)
完成时间为 11 小时

Introduction

...
Reading
1 个视频(共 8 分钟), 4 个阅读材料
Video1 个视频
Reading4 个阅读材料
Course Overview10分钟
Grading and Logistics10分钟
Suggested Readings分钟
About the Instructor10分钟
完成时间(小时)
完成时间为 11 小时

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 个视频(共 78 分钟), 1 个测验
Video7 个视频
Words9分钟
Permutations10分钟
k-permutations8分钟
Merry-go-rounds and Fermat’s little theorem 18分钟
Merry-go-rounds and Fermat’s little theorem 211分钟
Binomial coefficients14分钟
The Pascal triangle16分钟
Quiz1 个练习
Quiz 2分钟
2
完成时间(小时)
完成时间为 11 小时

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 个视频(共 87 分钟), 1 个测验
Video7 个视频
Balls in boxes and multisets 110分钟
Balls in boxes and multisets 26分钟
Integer compositions11分钟
Principle of inclusion and exclusion: two examples12分钟
Principle of inclusion and exclusion: general statement9分钟
The derangement problem19分钟
Quiz1 个练习
Quiz 3分钟
3
完成时间(小时)
完成时间为 14 小时

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 个视频(共 105 分钟), 1 个阅读材料, 1 个测验
Video11 个视频
Fibonacci numbers and the Pascal triangle7分钟
Domino tilings8分钟
Vending machine problem10分钟
Linear recurrence relations: definition7分钟
The characteristic equation8分钟
Linear recurrence relations of order 211分钟
The Binet formula11分钟
Sidebar: the golden ratio9分钟
Linear recurrence relations of arbitrary order8分钟
The case of roots with multiplicities12分钟
Reading1 个阅读材料
Spoilers! Solutions for quizzes 2, 3, and 4.分钟
Quiz1 个练习
Quiz 4分钟
4
完成时间(小时)
完成时间为 13 小时

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 个视频(共 73 分钟), 1 个阅读材料, 1 个测验
Video7 个视频
Recurrence relation for triangulations11分钟
The cashier problem9分钟
Dyck paths5分钟
Recurrence relations for Dyck paths9分钟
Reflection trick and a formula for Catalan numbers12分钟
Binary trees15分钟
Reading1 个阅读材料
Solutions10分钟
4.6
22 个审阅Chevron Right

热门审阅

创建者 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.

创建者 RRAug 22nd 2017

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

讲师

Avatar

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

关于 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...

常见问题

  • 注册以便获得证书后,您将有权访问所有视频、测验和编程作业(如果适用)。只有在您的班次开课之后,才可以提交和审阅同学互评作业。如果您选择在不购买的情况下浏览课程,可能无法访问某些作业。

  • 您购买证书后,将有权访问所有课程材料,包括评分作业。完成课程后,您的电子课程证书将添加到您的成就页中,您可以通过该页打印您的课程证书或将其添加到您的领英档案中。如果您只想阅读和查看课程内容,可以免费旁听课程。

还有其他问题吗?请访问 学生帮助中心