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中級

完成時間大約為35 小時

建議:8 weeks, 10-12 hours per week...

英語(English)

字幕:英語(English)

100% 在線

立即開始,按照自己的計劃學習。

可靈活調整截止日期

根據您的日程表重置截止日期。

中級

完成時間大約為35 小時

建議:8 weeks, 10-12 hours per week...

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 11 小時

Introduction

...
1 個視頻 (總計 8 分鐘), 4 個閱讀材料
1 個視頻
4 個閱讀材料
Course Overview10分鐘
Grading and Logistics10分鐘
Suggested Readings
About the Instructor10分鐘
完成時間為 11 小時

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 個視頻 (總計 78 分鐘), 1 個測驗
7 個視頻
Words9分鐘
Permutations10分鐘
k-permutations8分鐘
Merry-go-rounds and Fermat’s little theorem 18分鐘
Merry-go-rounds and Fermat’s little theorem 211分鐘
Binomial coefficients14分鐘
The Pascal triangle16分鐘
1 個練習
Quiz 2
2
完成時間為 11 小時

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 個視頻 (總計 87 分鐘), 1 個測驗
7 個視頻
Balls in boxes and multisets 110分鐘
Balls in boxes and multisets 26分鐘
Integer compositions11分鐘
Principle of inclusion and exclusion: two examples12分鐘
Principle of inclusion and exclusion: general statement9分鐘
The derangement problem19分鐘
1 個練習
Quiz 3
3
完成時間為 14 小時

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 個視頻 (總計 105 分鐘), 1 個閱讀材料, 1 個測驗
11 個視頻
Fibonacci numbers and the Pascal triangle7分鐘
Domino tilings8分鐘
Vending machine problem10分鐘
Linear recurrence relations: definition7分鐘
The characteristic equation8分鐘
Linear recurrence relations of order 211分鐘
The Binet formula11分鐘
Sidebar: the golden ratio9分鐘
Linear recurrence relations of arbitrary order8分鐘
The case of roots with multiplicities12分鐘
1 個閱讀材料
Spoilers! Solutions for quizzes 2, 3, and 4.
1 個練習
Quiz 4
4
完成時間為 13 小時

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 個視頻 (總計 73 分鐘), 1 個閱讀材料, 1 個測驗
7 個視頻
Recurrence relation for triangulations11分鐘
The cashier problem9分鐘
Dyck paths5分鐘
Recurrence relations for Dyck paths9分鐘
Reflection trick and a formula for Catalan numbers12分鐘
Binary trees15分鐘
1 個閱讀材料
Solutions10分鐘
5
完成時間為 12 小時

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 個視頻 (總計 87 分鐘), 1 個閱讀材料, 1 個測驗
9 個視頻
Formal power series11分鐘
When are formal power series invertible?9分鐘
Derivation of formal power series12分鐘
Binomial theorem for negative integer exponents8分鐘
Solving the Fibonacci recurrence relation9分鐘
Solving the Fibonacci recurrence 2: Binet formula6分鐘
Generating functions of linear recurrence relations are rational7分鐘
Solving linear recurrence relations: general case10分鐘
1 個閱讀材料
Math expressions10分鐘
1 個練習
Quiz 6
6
完成時間為 11 小時

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 個視頻 (總計 61 分鐘), 1 個測驗
6 個視頻
Derivation and integration of formal power series10分鐘
Chain rule. Inverse function theorem7分鐘
Logarithm. Logarithmic derivative5分鐘
Binomial theorem for arbitrary exponents13分鐘
Generating function for Catalan numbers14分鐘
1 個練習
Quiz 7
7
完成時間為 13 小時

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 個視頻 (總計 87 分鐘), 1 個閱讀材料, 1 個測驗
9 個視頻
Young diagrams4分鐘
Generating function for partitions15分鐘
Partitions with odd and distinct summands11分鐘
Sylvester’s bijection8分鐘
Euler’s pentagonal theorem12分鐘
Proof of Euler’s pentagonal theorem 18分鐘
Proof of Euler’s pentagonal theorem 214分鐘
Computing the number of partitions via the pentagonal theorem6分鐘
1 個閱讀材料
Spoilers! Solutions for quizzes 6, 7, and 8.
1 個練習
Quiz 8
8
完成時間為 14 小時

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 個視頻 (總計 80 分鐘), 1 個閱讀材料, 1 個測驗
8 個視頻
q-binomial coefficients: definition and first properties10分鐘
Recurrence relation for q-binomial coefficients 114分鐘
Recurrence relation for q-binomial coefficients 23分鐘
Explicit formula for q-binomial coefficients11分鐘
Euler’s partition function8分鐘
Sidebar: q-binomial coefficients in linear algebra9分鐘
q-binomial theorem10分鐘
1 個閱讀材料
Solutions10分鐘
4.6
25 個審閱Chevron Right

熱門審閱

創建者 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.

創建者 RRAug 22nd 2017

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

講師

Avatar

Evgeny Smirnov

Associate Professor
Faculty of Mathematics

關於 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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