Princeton University

Analytic Combinatorics

Taught in English

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Course

Gain insight into a topic and learn the fundamentals

Robert Sedgewick

Instructor: Robert Sedgewick

4.6

(63 reviews)

Intermediate level
Some related experience required
16 hours to complete
3 weeks at 5 hours a week
Flexible schedule
Learn at your own pace

Details to know

Assessments

8 quizzes

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There are 8 modules in this course

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.

What's included

7 videos2 readings1 quiz1 discussion prompt

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.

What's included

7 videos1 reading1 quiz1 discussion prompt

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.

What's included

5 videos1 reading1 quiz1 discussion prompt

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.

What's included

6 videos1 reading1 quiz1 discussion prompt

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.

What's included

6 videos1 reading1 quiz1 discussion prompt

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.

What's included

5 videos1 reading1 quiz1 discussion prompt

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.

What's included

6 videos1 reading1 quiz1 discussion prompt

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.

What's included

5 videos1 quiz

Instructor

Instructor ratings
5.0 (17 ratings)
Robert Sedgewick
Princeton University
7 Courses1,713,819 learners

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4.6

63 reviews

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Reviewed on Feb 15, 2020

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Reviewed on Aug 10, 2020

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