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
proposé par
Princeton University
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
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 vidéos (Total 73 min), 2 lectures, 1 quiz
7 vidéos
Brief History 9 min
Symbolic Method11 min
Trees and Strings 14 min
Powersets and Multisets 13 min
Compositions and Partitions 15 min
Substitution 6 min
Exercises 3 min
2 lectures
Getting Started10 min
Exercises from Lecture 110 min
1 exercice pour s'entraîner
Combinatorial Structures and OGFs4 min
Semaine
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 vidéos (Total 85 min), 1 lecture, 1 quiz
7 vidéos
Basics13 min
Symbolic Method for Labelled Classes 18 min
Words and Strings 12 min
Labelled trees 15 min
Mappings 17 min
Summary 4 min
Exercises 2 min
1 lecture
Exercises from Lecture 210 min
1 exercice pour s'entraîner
Labeled Structures and EGFs4 min
Semaine
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 vidéos (Total 84 min), 1 lecture, 1 quiz
5 vidéos
Basics19 min
Moment Calculations 24 min
OBGF examples 17 min
Labelled Classes 19 min
Exercises 2 min
1 lecture
Exercises from Lecture 310 min
1 exercice pour s'entraîner
Combinatorial Parameters and MGFs8 min
Semaine
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 vidéos (Total 109 min), 1 lecture, 1 quiz
6 vidéos
Roadmap13 min
Complex Functions 13 min
Rational Functions 19 min
Analytic Functions and Complex Integration 23 min
Meromorphic Functions 34 min
Exercises 3 min
1 lecture
Exercises from Lecture 410 min
1 exercice pour s'entraîner
Complex Analysis, Rational and Meromorphic Asymptotics4 min
Enseignant
Robert Sedgewick
William O. Baker *39 Professor of Computer ScienceComputer Science
À propos de 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.
Foire Aux Questions
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