À propos de ce cours
4.6
55 ratings
22 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

Niveau intermédiaire

Clock

Recommandé : 8 weeks, 10-12 hours per week

Approx. 34 heures pour terminer
Comment Dots

English

Sous-titres : English
Globe

Cours en ligne à 100 %

Commencez dès maintenant et apprenez aux horaires qui vous conviennent.
Calendar

Dates limites flexibles

Réinitialisez les dates limites selon votre disponibilité.
Intermediate Level

Niveau intermédiaire

Clock

Recommandé : 8 weeks, 10-12 hours per week

Approx. 34 heures pour terminer
Comment Dots

English

Sous-titres : English

Programme du cours : ce que vous apprendrez dans ce cours

1

Section
Clock
11 heures pour terminer

Introduction

...
Reading
1 vidéo (Total 8 min), 4 lectures
Video1 vidéo
Reading4 lectures
Course Overview10 min
Grading and Logistics10 min
Suggested Readings min
About the Instructor10 min
Clock
11 heures pour terminer

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 vidéos (Total 78 min), 1 quiz
Video7 vidéos
Words9 min
Permutations10 min
k-permutations8 min
Merry-go-rounds and Fermat’s little theorem 18 min
Merry-go-rounds and Fermat’s little theorem 211 min
Binomial coefficients14 min
The Pascal triangle16 min
Quiz1 exercice pour s'entraîner
Quiz 2 min

2

Section
Clock
11 heures pour terminer

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 vidéos (Total 87 min), 1 quiz
Video7 vidéos
Balls in boxes and multisets 110 min
Balls in boxes and multisets 26 min
Integer compositions11 min
Principle of inclusion and exclusion: two examples12 min
Principle of inclusion and exclusion: general statement9 min
The derangement problem19 min
Quiz1 exercice pour s'entraîner
Quiz 3 min

3

Section
Clock
14 heures pour terminer

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 vidéos (Total 105 min), 1 lecture, 1 quiz
Video11 vidéos
Fibonacci numbers and the Pascal triangle7 min
Domino tilings8 min
Vending machine problem10 min
Linear recurrence relations: definition7 min
The characteristic equation8 min
Linear recurrence relations of order 211 min
The Binet formula11 min
Sidebar: the golden ratio9 min
Linear recurrence relations of arbitrary order8 min
The case of roots with multiplicities12 min
Reading1 lecture
Spoilers! Solutions for quizzes 2, 3, and 4. min
Quiz1 exercice pour s'entraîner
Quiz 4 min

4

Section
Clock
13 heures pour terminer

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 vidéos (Total 73 min), 1 lecture, 1 quiz
Video7 vidéos
Recurrence relation for triangulations11 min
The cashier problem9 min
Dyck paths5 min
Recurrence relations for Dyck paths9 min
Reflection trick and a formula for Catalan numbers12 min
Binary trees15 min
Reading1 lecture
Solutions10 min
4.6

Meilleurs avis

par 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.

par RRAug 22nd 2017

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

Enseignant

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

À propos de 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...

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