4.6
60 个评分
23 个审阅

1

## Introduction

...
1 个视频 （总计 8 分钟）, 4 个阅读材料
1 个视频
4 个阅读材料
Course Overview10分钟

## 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....
7 个视频 （总计 78 分钟）, 1 个测验
7 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
1 个练习
Quiz 2
2

## 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)....
7 个视频 （总计 87 分钟）, 1 个测验
7 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
1 个练习
Quiz 3
3

## 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....
11 个视频 （总计 105 分钟）, 1 个阅读材料, 1 个测验
11 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
1 个阅读材料
Spoilers! Solutions for quizzes 2, 3, and 4.
1 个练习
Quiz 4
4

## 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....
7 个视频 （总计 73 分钟）, 1 个阅读材料, 1 个测验
7 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
1 个阅读材料
Solutions10分钟
4.6
23 个审阅

### 热门审阅

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.

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 国立高等经济大学

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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