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中級レベル

約35時間で修了

推奨:8 weeks, 10-12 hours per week...

英語

字幕:英語

100%オンライン

自分のスケジュールですぐに学習を始めてください。

柔軟性のある期限

スケジュールに従って期限をリセットします。

中級レベル

約35時間で修了

推奨:8 weeks, 10-12 hours per week...

英語

字幕:英語

シラバス - 本コースの学習内容

1
11時間で修了

Introduction

...
1件のビデオ (合計8分), 4 readings
1件のビデオ
4件の学習用教材
Course Overview10 分
Grading and Logistics10 分
Suggested Readings
About the Instructor10 分
11時間で修了

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件のビデオ (合計78分), 1 quiz
7件のビデオ
Words9 分
Permutations10 分
k-permutations8 分
Merry-go-rounds and Fermat’s little theorem 18 分
Merry-go-rounds and Fermat’s little theorem 211 分
Binomial coefficients14 分
The Pascal triangle16 分
1の練習問題
Quiz 2
2
11時間で修了

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件のビデオ (合計87分), 1 quiz
7件のビデオ
Balls in boxes and multisets 110 分
Balls in boxes and multisets 26 分
Integer compositions11 分
Principle of inclusion and exclusion: two examples12 分
Principle of inclusion and exclusion: general statement9 分
The derangement problem19 分
1の練習問題
Quiz 3
3
14時間で修了

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件のビデオ (合計105分), 1 reading, 1 quiz
11件のビデオ
Fibonacci numbers and the Pascal triangle7 分
Domino tilings8 分
Vending machine problem10 分
Linear recurrence relations: definition7 分
The characteristic equation8 分
Linear recurrence relations of order 211 分
The Binet formula11 分
Sidebar: the golden ratio9 分
Linear recurrence relations of arbitrary order8 分
The case of roots with multiplicities12 分
1件の学習用教材
Spoilers! Solutions for quizzes 2, 3, and 4.
1の練習問題
Quiz 4
4
13時間で修了

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件のビデオ (合計73分), 1 reading, 1 quiz
7件のビデオ
Recurrence relation for triangulations11 分
The cashier problem9 分
Dyck paths5 分
Recurrence relations for Dyck paths9 分
Reflection trick and a formula for Catalan numbers12 分
Binary trees15 分
1件の学習用教材
Solutions10 分
5
12時間で修了

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件のビデオ (合計87分), 1 reading, 1 quiz
9件のビデオ
Formal power series11 分
When are formal power series invertible?9 分
Derivation of formal power series12 分
Binomial theorem for negative integer exponents8 分
Solving the Fibonacci recurrence relation9 分
Solving the Fibonacci recurrence 2: Binet formula6 分
Generating functions of linear recurrence relations are rational7 分
Solving linear recurrence relations: general case10 分
1件の学習用教材
Math expressions10 分
1の練習問題
Quiz 6
6
11時間で修了

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件のビデオ (合計61分), 1 quiz
6件のビデオ
Derivation and integration of formal power series10 分
Chain rule. Inverse function theorem7 分
Logarithm. Logarithmic derivative5 分
Binomial theorem for arbitrary exponents13 分
Generating function for Catalan numbers14 分
1の練習問題
Quiz 7
7
13時間で修了

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件のビデオ (合計87分), 1 reading, 1 quiz
9件のビデオ
Young diagrams4 分
Generating function for partitions15 分
Partitions with odd and distinct summands11 分
Sylvester’s bijection8 分
Euler’s pentagonal theorem12 分
Proof of Euler’s pentagonal theorem 18 分
Proof of Euler’s pentagonal theorem 214 分
Computing the number of partitions via the pentagonal theorem6 分
1件の学習用教材
Spoilers! Solutions for quizzes 6, 7, and 8.
1の練習問題
Quiz 8
8
14時間で修了

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件のビデオ (合計80分), 1 reading, 1 quiz
8件のビデオ
q-binomial coefficients: definition and first properties10 分
Recurrence relation for q-binomial coefficients 114 分
Recurrence relation for q-binomial coefficients 23 分
Explicit formula for q-binomial coefficients11 分
Euler’s partition function8 分
Sidebar: q-binomial coefficients in linear algebra9 分
q-binomial theorem10 分
1件の学習用教材
Solutions10 分
4.6
25件のレビューChevron Right

人気のレビュー

by 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.

by RRAug 22nd 2017

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

講師

Avatar

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

ロシア国立研究大学経済高等学院(National Research University Higher School of Economics)について

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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