课程信息
4,300

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中级

完成时间大约为20 小时

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

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中级

完成时间大约为20 小时

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 2 小时

Combinatorial Structures and OGFs

Our first lecture is about the symbolic method, where we define combinatorial constructions that we can use to define classes of combinatorial objects. The constructions are integrated with transfer theorems that lead to equations that define generating functions whose coefficients enumerate the classes. We consider numerous examples from classical combinatorics. ...
7 个视频 (总计 73 分钟), 2 个阅读材料, 1 个测验
7 个视频
Symbolic Method11分钟
Trees and Strings 14分钟
Powersets and Multisets 13分钟
Compositions and Partitions 15分钟
Substitution 6分钟
Exercises 3分钟
2 个阅读材料
Getting Started10分钟
Exercises from Lecture 110分钟
1 个练习
Combinatorial Structures and OGFs4分钟
2
完成时间为 2 小时

Labelled Structures and EGFs

This lecture introduces labelled objects, where the atoms that we use to build objects are distinguishable. We use exponential generating functions EGFs to study combinatorial classes built from labelled objects. As in Lecture 1, we define combinatorial constructions that lead to EGF equations, and consider numerous examples from classical combinatorics. ...
7 个视频 (总计 85 分钟), 1 个阅读材料, 1 个测验
7 个视频
Basics13分钟
Symbolic Method for Labelled Classes 18分钟
Words and Strings 12分钟
Labelled trees 15分钟
Mappings 17分钟
Summary 4分钟
Exercises 2分钟
1 个阅读材料
Exercises from Lecture 210分钟
1 个练习
Labeled Structures and EGFs4分钟
3
完成时间为 2 小时

Combinatorial Parameters and MGFs

This lecture describes the process of adding variables to mark parameters and then using the constructions form Lectures 1 and 2 and natural extensions of the transfer theorems to define multivariate GFs that contain information about parameters. We concentrate on bivariate generating functions (BGFs), where one variable marks the size of an object and the other marks the value of a parameter. After studying ways of computing the mean, standard deviation and other moments from BGFs, we consider several examples in some detail. ...
5 个视频 (总计 84 分钟), 1 个阅读材料, 1 个测验
5 个视频
Basics19分钟
Moment Calculations 24分钟
OBGF examples 17分钟
Labelled Classes 19分钟
Exercises 2分钟
1 个阅读材料
Exercises from Lecture 310分钟
1 个练习
Combinatorial Parameters and MGFs8分钟
4
完成时间为 2 小时

Complex Analysis, Rational and Meromorphic Asymptotics

This week we introduce the idea of viewing generating functions as analytic objects, which leads us to asymptotic estimates of coefficients. The approach is most fruitful when we consider GFs as complex functions, so we introduce and apply basic concepts in complex analysis. We start from basic principles, so prior knowledge of complex analysis is not required. ...
6 个视频 (总计 109 分钟), 1 个阅读材料, 1 个测验
6 个视频
Roadmap13分钟
Complex Functions 13分钟
Rational Functions 19分钟
Analytic Functions and Complex Integration 23分钟
Meromorphic Functions 34分钟
Exercises 3分钟
1 个阅读材料
Exercises from Lecture 410分钟
1 个练习
Complex Analysis, Rational and Meromorphic Asymptotics4分钟
5
完成时间为 1 小时

Applications of Rational and Meromorphic Asymptotics

We consider applications of the general transfer theorem of the previous lecture to many of the classic combinatorial classes that we encountered in Lectures 1 and 2. Then we consider a universal law that gives asymptotics for a broad swath of combinatorial classes built with the sequence construction....
6 个视频 (总计 63 分钟), 1 个阅读材料, 1 个测验
6 个视频
Bitstrings16分钟
Other Familiar Examples 15分钟
Restricted Compositions 9分钟
Supercritical Sequence Schema 14分钟
Summary 3分钟
Exercises 2分钟
1 个阅读材料
Exercises from Lecture 510分钟
1 个练习
Applications of Complex Analysis, Rational and Meromorphic Asymptotics4分钟
6
完成时间为 2 小时

Singularity Analysis

This lecture addresses the basic Flajolet-Odlyzko theorem, where we find the domain of analyticity of the function near its dominant singularity, approximate using functions from standard scale, and then transfer to coefficient asymptotics term-by-term....
5 个视频 (总计 82 分钟), 1 个阅读材料, 1 个测验
5 个视频
Prelude22分钟
Standard Function Scale 11分钟
Singularity Analysis 20分钟
Schemas and Transfer Theorems 25分钟
Exercises 2分钟
1 个阅读材料
Exercises from Lecture 610分钟
1 个练习
Singularity Analysis of Generating Functions4分钟
7
完成时间为 1 小时

Applications of Singularity Analysis

We see how the Flajolet-Odlyzko approach leads to universal laws covering combinatorial classes built with the set, multiset, and recursive sequence constructions. Then we consider applications to many of the classic combinatorial classes that we encountered in Lectures 1 and 2....
6 个视频 (总计 65 分钟), 1 个阅读材料, 1 个测验
6 个视频
Labelled Sets 15分钟
Mappings 11分钟
Tree-like Classes 16分钟
Summary 3分钟
Exercises 2分钟
1 个阅读材料
Exercises from Lecture 710分钟
1 个练习
Applications of Singularity Analysis4分钟
8
完成时间为 1 小时

Saddle Point Asymptotics

We consider the saddle point method, a general technique for contour integration that also provides an effective path to the development of coefficient asymptotics for GFs with no singularities. As usual, we consider the application of this method to several of the classic problems introduced in Lectures 1 and 2....
5 个视频 (总计 63 分钟), 1 个测验
5 个视频
Saddle Point Bounds 9分钟
Saddle Point Asymptotics 18分钟
Applications 5分钟
AC Wrap-up 11分钟
1 个练习
Saddle-Point Asymptotics6分钟

讲师

Avatar

Robert Sedgewick

William O. Baker *39 Professor of Computer Science
Computer Science

关于 普林斯顿大学

Princeton University is a private research university located in Princeton, New Jersey, United States. It is one of the eight universities of the Ivy League, and one of the nine Colonial Colleges founded before the American Revolution....

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