课程信息
4.7
51 ratings
21 reviews
Enumerative combinatorics deals with finite sets and their cardinalities. In other words, a typical problem of enumerative combinatorics is to find the number of ways a certain pattern can be formed. In the first part of our course we will be dealing with elementary combinatorial objects and notions: permutations, combinations, compositions, Fibonacci and Catalan numbers etc. In the second part of the course we introduce the notion of generating functions and use it to study recurrence relations and partition numbers. The course is mostly self-contained. However, some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful....
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Intermediate Level

中级

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建议:8 weeks, 10-12 hours per week

完成时间大约为34 小时
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English

字幕:English
Globe

100% 在线课程

立即开始,按照自己的计划学习。
Calendar

可灵活调整截止日期

根据您的日程表重置截止日期。
Intermediate Level

中级

Clock

建议:8 weeks, 10-12 hours per week

完成时间大约为34 小时
Comment Dots

English

字幕:English

教学大纲 - 您将从这门课程中学到什么

1

章节
Clock
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
Clock
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

章节
Clock
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

章节
Clock
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

章节
Clock
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
4.7

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

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...

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