About this Course
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.
Globe

100% online course

Start instantly and learn at your own schedule.
Intermediate Level

Intermediate Level

Clock

Approx. 35 hours to complete

Suggested: 8 weeks, 10-12 hours per week
Comment Dots

English

Subtitles: English
Globe

100% online course

Start instantly and learn at your own schedule.
Intermediate Level

Intermediate Level

Clock

Approx. 35 hours to complete

Suggested: 8 weeks, 10-12 hours per week
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 videos
Reading4 readings
Course Overview10m
Grading and Logistics10m
Suggested Readings0m
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 exercises
Quiz 20m

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 exercises
Quiz 30m

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 readings
Spoilers! Solutions for quizzes 2, 3, and 4.0m
Quiz1 practice exercises
Quiz 40m

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 readings
Solutions10m

5

Section
Clock
12 hours to complete

Generating functions: a unified approach to combinatorial problems. Solving linear recurrences

We introduce the central notion of our course, the notion of a generating function. We start with studying properties of formal power series and then apply the machinery of generating functions to solving linear recurrence relations....
Reading
9 videos (Total 87 min), 1 reading, 1 quiz
Video9 videos
Formal power series11m
When are formal power series invertible?9m
Derivation of formal power series12m
Binomial theorem for negative integer exponents8m
Solving the Fibonacci recurrence relation9m
Solving the Fibonacci recurrence 2: Binet formula6m
Generating functions of linear recurrence relations are rational7m
Solving linear recurrence relations: general case10m
Reading1 readings
Math expressions10m
Quiz1 practice exercises
Quiz 60m

6

Section
Clock
11 hours to complete

Generating functions, continued. Generating function of the Catalan sequence

In this lecture we discuss further properties of formal power series. In particular, we prove an analogue of the binomial theorem for an arbitrary rational exponent. We apply this technique to computing the generating function of the sequence of Catalan numbers....
Reading
6 videos (Total 61 min), 1 quiz
Video6 videos
Derivation and integration of formal power series10m
Chain rule. Inverse function theorem7m
Logarithm. Logarithmic derivative5m
Binomial theorem for arbitrary exponents13m
Generating function for Catalan numbers14m
Quiz1 practice exercises
Quiz 70m

7

Section
Clock
13 hours to complete

Partitions. Euler’s generating function for partitions and pentagonal formula

In this lecture we introduce partitions, i.e. the number of ways to present a given integer as a sum of ordered integer summands. There is no closed formula for the number of partitions; however, it is possible to compute their generating function. We study the properties of this generating function, including the famous Pentagonal theorem, due to Leonhard Euler....
Reading
9 videos (Total 87 min), 1 reading, 1 quiz
Video9 videos
Young diagrams4m
Generating function for partitions15m
Partitions with odd and distinct summands11m
Sylvester’s bijection8m
Euler’s pentagonal theorem12m
Proof of Euler’s pentagonal theorem 18m
Proof of Euler’s pentagonal theorem 214m
Computing the number of partitions via the pentagonal theorem6m
Reading1 readings
Spoilers! Solutions for quizzes 6, 7, and 8.0m
Quiz1 practice exercises
Quiz 80m

8

Section
Clock
14 hours to complete

Gaussian binomial coefficients. “Quantum” versions of combinatorial identities

Our final lecture is devoted to the so-called "q-analogues" of various combinatorial notions and identities. As a general principle, we replace identities with numbers by identities with polynomials in a certain variable, usually denoted by q, that return the original statement as q tends to 1. This approach turns out to be extremely useful in various branches of mathematics, from number theory to representation theory....
Reading
8 videos (Total 80 min), 1 reading, 1 quiz
Video8 videos
q-binomial coefficients: definition and first properties10m
Recurrence relation for q-binomial coefficients 114m
Recurrence relation for q-binomial coefficients 23m
Explicit formula for q-binomial coefficients11m
Euler’s partition function8m
Sidebar: q-binomial coefficients in linear algebra9m
q-binomial theorem10m
Reading1 readings
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

Avatar

Evgeny Smirnov

Associate Professor

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.

  • If you pay for this course, you will have access to all of the features and content you need to earn a Course Certificate. If you complete the course successfully, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Note that the Course Certificate does not represent official academic credit from the partner institution offering the course.

  • Yes! Coursera provides financial aid to learners who would like to complete a course but cannot afford the course fee. To apply for aid, select "Learn more and apply" in the Financial Aid section below the "Enroll" button. You'll be prompted to complete a simple application; no other paperwork is required.

More questions? Visit the Learner Help Center