About this Course
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

Intermediate Level

Clock

Suggested: 8 weeks, 10-12 hours per week

Approx. 34 hours to complete
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English

Subtitles: English
Globe

100% online courses

Start instantly and learn at your own schedule.
Calendar

Flexible deadlines

Reset deadlines in accordance to your schedule.
Intermediate Level

Intermediate Level

Clock

Suggested: 8 weeks, 10-12 hours per week

Approx. 34 hours to complete
Comment Dots

English

Subtitles: English

Syllabus - What you will learn from this course

1

Section
Clock
11 hours to complete

Introduction

...
Reading
1 video (Total 8 min), 4 readings
Video1 video
Reading4 readings
Course Overview10m
Grading and Logistics10m
Suggested Readingsm
About the Instructor10m
Clock
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....
Reading
7 videos (Total 78 min), 1 quiz
Video7 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
Quiz1 practice exercise
Quiz 2m

2

Section
Clock
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)....
Reading
7 videos (Total 87 min), 1 quiz
Video7 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
Quiz1 practice exercise
Quiz 3m

3

Section
Clock
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....
Reading
11 videos (Total 105 min), 1 reading, 1 quiz
Video11 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
Reading1 reading
Spoilers! Solutions for quizzes 2, 3, and 4.m
Quiz1 practice exercise
Quiz 4m

4

Section
Clock
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....
Reading
7 videos (Total 73 min), 1 reading, 1 quiz
Video7 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
Reading1 reading
Solutions10m
4.7

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

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