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National Research University Higher School of Economics

课程信息

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....

立即开始，按照自己的计划学习。

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完成时间大约为34 小时

字幕：English

立即开始，按照自己的计划学习。

根据您的日程表重置截止日期。

完成时间大约为34 小时

字幕：English

章节

...

1 个视频（共 8 分钟）, 4 个阅读材料

Introduction7分钟

Course Overview10分钟

Grading and Logistics10分钟

Suggested Readings分钟

About the Instructor10分钟

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....

7 个视频（共 78 分钟）, 1 个测验

Words9分钟

Permutations10分钟

k-permutations8分钟

Merry-go-rounds and Fermat’s little theorem 18分钟

Merry-go-rounds and Fermat’s little theorem 211分钟

Binomial coefficients14分钟

The Pascal triangle16分钟

Quiz 2分钟

章节

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)....

7 个视频（共 87 分钟）, 1 个测验

Balls in boxes and multisets 110分钟

Balls in boxes and multisets 26分钟

Integer compositions11分钟

Principle of inclusion and exclusion: two examples12分钟

Principle of inclusion and exclusion: general statement9分钟

The derangement problem19分钟

Quiz 3分钟

章节

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....

11 个视频（共 105 分钟）, 1 个阅读材料, 1 个测验

Fibonacci numbers and the Pascal triangle7分钟

Domino tilings8分钟

Vending machine problem10分钟

Linear recurrence relations: definition7分钟

The characteristic equation8分钟

Linear recurrence relations of order 211分钟

The Binet formula11分钟

Sidebar: the golden ratio9分钟

Linear recurrence relations of arbitrary order8分钟

The case of roots with multiplicities12分钟

Spoilers! Solutions for quizzes 2, 3, and 4.分钟

Quiz 4分钟

章节

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....

7 个视频（共 73 分钟）, 1 个阅读材料, 1 个测验

Recurrence relation for triangulations11分钟

The cashier problem9分钟

Dyck paths5分钟

Recurrence relations for Dyck paths9分钟

Reflection trick and a formula for Catalan numbers12分钟

Binary trees15分钟

Solutions10分钟

4.7

创建者 RA•Mar 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.

创建者 RR•Aug 22nd 2017

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

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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