Об этом курсе
4.7
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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

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Предполагаемая нагрузка: 8 weeks, 10-12 hours per week

Прибл. 34 ч. на завершение
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English

Субтитры: English
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Только онлайн-курсы

Начните сейчас и учитесь по собственному графику.
Calendar

Гибкие сроки

Назначьте сроки сдачи в соответствии со своим графиком.
Intermediate Level

Промежуточный уровень

Clock

Предполагаемая нагрузка: 8 weeks, 10-12 hours per week

Прибл. 34 ч. на завершение
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English

Субтитры: English

Программа курса: что вы изучите

1

Раздел
Clock
11 ч. на завершение

Introduction

...
Reading
1 видео (всего 8 мин.), 4 материалов для самостоятельного изучения
Video1 видео
Reading4 материала для самостоятельного изучения
Course Overview10мин
Grading and Logistics10мин
Suggested Readingsмин
About the Instructor10мин
Clock
11 ч. на завершение

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 видео (всего 78 мин.), 1 тест
Video7 видео
Words9мин
Permutations10мин
k-permutations8мин
Merry-go-rounds and Fermat’s little theorem 18мин
Merry-go-rounds and Fermat’s little theorem 211мин
Binomial coefficients14мин
The Pascal triangle16мин
Quiz1 практическое упражнение
Quiz 2мин

2

Раздел
Clock
11 ч. на завершение

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 видео (всего 87 мин.), 1 тест
Video7 видео
Balls in boxes and multisets 110мин
Balls in boxes and multisets 26мин
Integer compositions11мин
Principle of inclusion and exclusion: two examples12мин
Principle of inclusion and exclusion: general statement9мин
The derangement problem19мин
Quiz1 практическое упражнение
Quiz 3мин

3

Раздел
Clock
14 ч. на завершение

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 видео (всего 105 мин.), 1 материал для самостоятельного изучения, 1 тест
Video11 видео
Fibonacci numbers and the Pascal triangle7мин
Domino tilings8мин
Vending machine problem10мин
Linear recurrence relations: definition7мин
The characteristic equation8мин
Linear recurrence relations of order 211мин
The Binet formula11мин
Sidebar: the golden ratio9мин
Linear recurrence relations of arbitrary order8мин
The case of roots with multiplicities12мин
Reading1 материал для самостоятельного изучения
Spoilers! Solutions for quizzes 2, 3, and 4.мин
Quiz1 практическое упражнение
Quiz 4мин

4

Раздел
Clock
13 ч. на завершение

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 видео (всего 73 мин.), 1 материал для самостоятельного изучения, 1 тест
Video7 видео
Recurrence relation for triangulations11мин
The cashier problem9мин
Dyck paths5мин
Recurrence relations for Dyck paths9мин
Reflection trick and a formula for Catalan numbers12мин
Binary trees15мин
Reading1 материал для самостоятельного изучения
Solutions10мин
4.7

Лучшие рецензии

автор: 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.

автор: RRAug 22nd 2017

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

Преподаватель

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

О 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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