4.6
55 ratings
22 reviews

#### 100% online

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

#### Approx. 35 hours to complete

Suggested: 8 weeks, 10-12 hours per week...

#### English

Subtitles: English

#### 100% online

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

#### Approx. 35 hours to complete

Suggested: 8 weeks, 10-12 hours per week...

#### English

Subtitles: English

### Syllabus - What you will learn from this course

Week
1
11 hours to complete

## Introduction

...
1 video (Total 8 min), 4 readings
1 video
Course Overview10m
11 hours to complete

## 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 videos (Total 78 min), 1 quiz
7 videos
Permutations10m
k-permutations8m
Merry-go-rounds and Fermat’s little theorem 18m
Merry-go-rounds and Fermat’s little theorem 211m
Binomial coefficients14m
The Pascal triangle16m
1 practice exercise
Quiz 2m
Week
2
11 hours to complete

## 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 videos (Total 87 min), 1 quiz
7 videos
Balls in boxes and multisets 110m
Balls in boxes and multisets 26m
Integer compositions11m
Principle of inclusion and exclusion: two examples12m
Principle of inclusion and exclusion: general statement9m
The derangement problem19m
1 practice exercise
Quiz 3m
Week
3
14 hours to complete

## 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 videos (Total 105 min), 1 reading, 1 quiz
11 videos
Fibonacci numbers and the Pascal triangle7m
Domino tilings8m
Vending machine problem10m
Linear recurrence relations: definition7m
The characteristic equation8m
Linear recurrence relations of order 211m
The Binet formula11m
Sidebar: the golden ratio9m
Linear recurrence relations of arbitrary order8m
The case of roots with multiplicities12m
Spoilers! Solutions for quizzes 2, 3, and 4.m
1 practice exercise
Quiz 4m
Week
4
13 hours to complete

## 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 videos (Total 73 min), 1 reading, 1 quiz
7 videos
Recurrence relation for triangulations11m
The cashier problem9m
Dyck paths5m
Recurrence relations for Dyck paths9m
Reflection trick and a formula for Catalan numbers12m
Binary trees15m
Solutions10m
4.6
22 Reviews

### Top Reviews

By 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.

By RRAug 22nd 2017

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

## Instructor

### Evgeny Smirnov

Associate Professor
Faculty of Mathematics

## About 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...

## Frequently Asked Questions

• Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.