Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. This course introduces the symbolic method to derive functional relations among ordinary, exponential, and multivariate generating functions, and methods in complex analysis for deriving accurate asymptotics from the GF equations.
All the features of this course are available for free. It does not offer a certificate upon completion.
Analytic Combinatorics
oferecido por
Princeton University
Programa - O que você aprenderá com este curso
Semana
1Combinatorial 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 vídeos (total de (Total 73 mín.) min), 2 leituras, 1 teste
7 videos
Brief History 9min
Symbolic Method11min
Trees and Strings 14min
Powersets and Multisets 13min
Compositions and Partitions 15min
Substitution 6min
Exercises 3min
2 leituras
Getting Started10min
Exercises from Lecture 110min
1 exercício prático
Combinatorial Structures and OGFs4min
Semana
2Labelled 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 vídeos (total de (Total 85 mín.) min), 1 leitura, 1 teste
7 videos
Basics13min
Symbolic Method for Labelled Classes 18min
Words and Strings 12min
Labelled trees 15min
Mappings 17min
Summary 4min
Exercises 2min
1 leituras
Exercises from Lecture 210min
1 exercício prático
Labeled Structures and EGFs4min
Semana
3Combinatorial 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 vídeos (total de (Total 84 mín.) min), 1 leitura, 1 teste
1 leituras
Exercises from Lecture 310min
1 exercício prático
Combinatorial Parameters and MGFs8min
Semana
4Complex 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 vídeos (total de (Total 109 mín.) min), 1 leitura, 1 teste
6 videos
Roadmap13min
Complex Functions 13min
Rational Functions 19min
Analytic Functions and Complex Integration 23min
Meromorphic Functions 34min
Exercises 3min
1 leituras
Exercises from Lecture 410min
1 exercício prático
Complex Analysis, Rational and Meromorphic Asymptotics4min
Instrutores
Robert Sedgewick
William O. Baker *39 Professor of Computer ScienceComputer Science
Sobre Princeton University
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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