Combinatorics is a branch of mathematics which deals with finite sets and their cardinalities. It is about counting certain objects having certain given properties. A typical question in combinatorics starts with the words "In how many ways" (we can do something).

Offered By

## Introduction to Enumerative Combinatorics

HSE University## About this Course

Some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.

Some acquaintance with basic linear algebra and analysis (including Taylor series expansion) may be very helpful.

## Offered by

### HSE University

HSE University 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 communicamathematics, engineering, and more.

## Syllabus - What you will learn from this course

**11 hours to complete**

## Introduction

**11 hours to complete**

**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.

**11 hours to complete**

**12 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).

**12 hours to complete**

**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.

**14 hours to complete**

**14 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.

**14 hours to complete**

## Reviews

### TOP REVIEWS FROM INTRODUCTION TO ENUMERATIVE COMBINATORICS

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.

very nice course.very well taught by professor.one thing that can be improved is detailed solution of quizzes and assignments.thanks for the course:)

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

This course is very well put together. The lectures are very clear and cover a wide range of interesting topics in combinatorics.

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