课程信息
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中级

完成时间大约为35 小时

建议:8 weeks, 10-12 hours per week...

英语(English)

字幕:英语(English)

100% 在线

立即开始,按照自己的计划学习。

可灵活调整截止日期

根据您的日程表重置截止日期。

中级

完成时间大约为35 小时

建议:8 weeks, 10-12 hours per week...

英语(English)

字幕:英语(English)

教学大纲 - 您将从这门课程中学到什么

1
完成时间为 11 小时

Introduction

...
1 个视频 (总计 8 分钟), 4 个阅读材料
1 个视频
4 个阅读材料
Course Overview10分钟
Grading and Logistics10分钟
Suggested Readings
About the Instructor10分钟
完成时间为 11 小时

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....
7 个视频 (总计 78 分钟), 1 个测验
7 个视频
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
1 个练习
Quiz 2
2
完成时间为 11 小时

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)....
7 个视频 (总计 87 分钟), 1 个测验
7 个视频
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
1 个练习
Quiz 3
3
完成时间为 14 小时

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....
11 个视频 (总计 105 分钟), 1 个阅读材料, 1 个测验
11 个视频
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
1 个阅读材料
Spoilers! Solutions for quizzes 2, 3, and 4.
1 个练习
Quiz 4
4
完成时间为 13 小时

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....
7 个视频 (总计 73 分钟), 1 个阅读材料, 1 个测验
7 个视频
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
1 个阅读材料
Solutions10分钟
5
完成时间为 12 小时

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....
9 个视频 (总计 87 分钟), 1 个阅读材料, 1 个测验
9 个视频
Formal power series11分钟
When are formal power series invertible?9分钟
Derivation of formal power series12分钟
Binomial theorem for negative integer exponents8分钟
Solving the Fibonacci recurrence relation9分钟
Solving the Fibonacci recurrence 2: Binet formula6分钟
Generating functions of linear recurrence relations are rational7分钟
Solving linear recurrence relations: general case10分钟
1 个阅读材料
Math expressions10分钟
1 个练习
Quiz 6
6
完成时间为 11 小时

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....
6 个视频 (总计 61 分钟), 1 个测验
6 个视频
Derivation and integration of formal power series10分钟
Chain rule. Inverse function theorem7分钟
Logarithm. Logarithmic derivative5分钟
Binomial theorem for arbitrary exponents13分钟
Generating function for Catalan numbers14分钟
1 个练习
Quiz 7
7
完成时间为 13 小时

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....
9 个视频 (总计 87 分钟), 1 个阅读材料, 1 个测验
9 个视频
Young diagrams4分钟
Generating function for partitions15分钟
Partitions with odd and distinct summands11分钟
Sylvester’s bijection8分钟
Euler’s pentagonal theorem12分钟
Proof of Euler’s pentagonal theorem 18分钟
Proof of Euler’s pentagonal theorem 214分钟
Computing the number of partitions via the pentagonal theorem6分钟
1 个阅读材料
Spoilers! Solutions for quizzes 6, 7, and 8.
1 个练习
Quiz 8
8
完成时间为 14 小时

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....
8 个视频 (总计 80 分钟), 1 个阅读材料, 1 个测验
8 个视频
q-binomial coefficients: definition and first properties10分钟
Recurrence relation for q-binomial coefficients 114分钟
Recurrence relation for q-binomial coefficients 23分钟
Explicit formula for q-binomial coefficients11分钟
Euler’s partition function8分钟
Sidebar: q-binomial coefficients in linear algebra9分钟
q-binomial theorem10分钟
1 个阅读材料
Solutions10分钟
4.6
25 个审阅Chevron Right

热门审阅

创建者 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.

创建者 RRAug 22nd 2017

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

讲师

Avatar

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

关于 国立高等经济大学

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 communicamathematics, engineering, and more. Learn more on www.hse.ru...

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