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完成時間大約為20 小時

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中級

完成時間大約為20 小時

英語(English)

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教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 2 小時

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 個視頻 (總計 73 分鐘), 2 個閱讀材料, 1 個測驗
7 個視頻
Symbolic Method11分鐘
Trees and Strings 14分鐘
Powersets and Multisets 13分鐘
Compositions and Partitions 15分鐘
Substitution 6分鐘
Exercises 3分鐘
2 個閱讀材料
Getting Started10分鐘
Exercises from Lecture 110分鐘
1 個練習
Combinatorial Structures and OGFs4分鐘
2
完成時間為 2 小時

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 個視頻 (總計 85 分鐘), 1 個閱讀材料, 1 個測驗
7 個視頻
Basics13分鐘
Symbolic Method for Labelled Classes 18分鐘
Words and Strings 12分鐘
Labelled trees 15分鐘
Mappings 17分鐘
Summary 4分鐘
Exercises 2分鐘
1 個閱讀材料
Exercises from Lecture 210分鐘
1 個練習
Labeled Structures and EGFs4分鐘
3
完成時間為 2 小時

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 個視頻 (總計 84 分鐘), 1 個閱讀材料, 1 個測驗
5 個視頻
Basics19分鐘
Moment Calculations 24分鐘
OBGF examples 17分鐘
Labelled Classes 19分鐘
Exercises 2分鐘
1 個閱讀材料
Exercises from Lecture 310分鐘
1 個練習
Combinatorial Parameters and MGFs8分鐘
4
完成時間為 2 小時

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 個視頻 (總計 109 分鐘), 1 個閱讀材料, 1 個測驗
6 個視頻
Roadmap13分鐘
Complex Functions 13分鐘
Rational Functions 19分鐘
Analytic Functions and Complex Integration 23分鐘
Meromorphic Functions 34分鐘
Exercises 3分鐘
1 個閱讀材料
Exercises from Lecture 410分鐘
1 個練習
Complex Analysis, Rational and Meromorphic Asymptotics4分鐘
5
完成時間為 1 小時

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 個視頻 (總計 63 分鐘), 1 個閱讀材料, 1 個測驗
6 個視頻
Bitstrings16分鐘
Other Familiar Examples 15分鐘
Restricted Compositions 9分鐘
Supercritical Sequence Schema 14分鐘
Summary 3分鐘
Exercises 2分鐘
1 個閱讀材料
Exercises from Lecture 510分鐘
1 個練習
Applications of Complex Analysis, Rational and Meromorphic Asymptotics4分鐘
6
完成時間為 2 小時

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 個視頻 (總計 82 分鐘), 1 個閱讀材料, 1 個測驗
5 個視頻
Prelude22分鐘
Standard Function Scale 11分鐘
Singularity Analysis 20分鐘
Schemas and Transfer Theorems 25分鐘
Exercises 2分鐘
1 個閱讀材料
Exercises from Lecture 610分鐘
1 個練習
Singularity Analysis of Generating Functions4分鐘
7
完成時間為 1 小時

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 個視頻 (總計 65 分鐘), 1 個閱讀材料, 1 個測驗
6 個視頻
Labelled Sets 15分鐘
Mappings 11分鐘
Tree-like Classes 16分鐘
Summary 3分鐘
Exercises 2分鐘
1 個閱讀材料
Exercises from Lecture 710分鐘
1 個練習
Applications of Singularity Analysis4分鐘
8
完成時間為 1 小時

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 個視頻 (總計 63 分鐘), 1 個測驗
5 個視頻
Saddle Point Bounds 9分鐘
Saddle Point Asymptotics 18分鐘
Applications 5分鐘
AC Wrap-up 11分鐘
1 個練習
Saddle-Point Asymptotics6分鐘

講師

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Robert Sedgewick

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

關於 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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