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. It is one of the eight universities of the Ivy League, and one of the nine Colonial Colleges founded before the American Revolution. Our first lecture is about the symbolic method, where we define combinatorial constructions that we can use to define classes of combinatorial objects.

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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. It is one of the eight universities of the Ivy League, and one of the nine Colonial Colleges founded before the American Revolution. 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. 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.

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. 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. 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. 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.

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. 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.

Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments. This Course doesn't carry university credit, but some universities may choose to accept Course Certificates for credit. Check with your institution to learn more. More questions?

Visit the Learner Help Center. Loupe Copy. Math and Logic. Analytic Combinatorics. Robert Sedgewick. Offered By. Analytic Combinatorics Princeton University. About this Course 12, recent views. Flexible deadlines. Flexible deadlines Reset deadlines in accordance to your schedule.

Intermediate Level. Hours to complete. Available languages. English Subtitles: English. Robert Sedgewick William O. Offered by. Syllabus - What you will learn from this course. Week 1. Video 7 videos. Brief History 9m. Symbolic Method 11m. Trees and Strings 14m. Powersets and Multisets 13m. Compositions and Partitions 15m. Substitution 6m. Exercises 3m. Reading 2 readings. Getting Started 10m. Exercises from Lecture 1 10m.

Quiz 1 practice exercise. Combinatorial Structures and OGFs 4m. Week 2. Basics 13m. Symbolic Method for Labelled Classes 18m. Words and Strings 12m. Labelled trees 15m. Mappings 17m. Summary 4m. Exercises 2m.

Reading 1 reading. Exercises from Lecture 2 10m. Labeled Structures and EGFs 4m. Week 3. Video 5 videos. Basics 19m. Moment Calculations 24m. OBGF examples 17m. Labelled Classes 19m. Exercises from Lecture 3 10m.

Combinatorial Parameters and MGFs 8m. Week 4. Video 6 videos. Roadmap 13m. Complex Functions 13m. Rational Functions 19m. Analytic Functions and Complex Integration 23m. Meromorphic Functions 34m. Exercises from Lecture 4 10m. Complex Analysis, Rational and Meromorphic Asymptotics 4m. Show More. Week 5. Bitstrings 16m. Other Familiar Examples 15m. Restricted Compositions 9m. Supercritical Sequence Schema 14m. Summary 3m. Exercises from Lecture 5 10m. Week 6.

Prelude 22m. Standard Function Scale 11m.

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## Symbolic method (combinatorics)

Analytic Combinatorics "If you can specify it, you can analyze it. Flajolet Online course materials. Click here for access to studio-produced lecture videos and associated lecture slides that provide an introduction to analytic combinatorics. Analytic combinatorics is a branch of mathematics that aims to enable precise quantitative predictions of the properties of large combinatorial structures, by connecting via generating functions formal descriptions of combinatorial structures with methods from complex and asymptotic analysis. The textbook Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick is the definitive treatment of the topic. The full text of the book is available for download here and you can purchase a hardcopy at Amazon or Cambridge University Press. Chapter 1: Combinatorial Structures and Ordinary Generating Functions introduces the symbolic method, where we define combinatorial constructions that we can use to define classes of combinatorial objects.

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## Analytic Combinatorics

Analytic Combinatorics is a book on the mathematics of combinatorial enumeration , using generating functions and complex analysis to understand the growth rates of the numbers of combinatorial objects. It won the Leroy P. Steele Prize in The main part of the book is organized into three parts.

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