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#### English

Subtitles: English

#### 100% online

Start instantly and learn at your own schedule.

#### English

Subtitles: English

### Syllabus - What you will learn from this course

Week
1
2 hours to complete

## 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 videos (Total 73 min), 2 readings, 1 quiz
7 videos
Symbolic Method11m
Trees and Strings 14m
Powersets and Multisets 13m
Compositions and Partitions 15m
Substitution 6m
Exercises 3m
Getting Started10m
Exercises from Lecture 110m
1 practice exercise
Combinatorial Structures and OGFs4m
Week
2
2 hours to complete

## 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 videos (Total 85 min), 1 reading, 1 quiz
7 videos
Symbolic Method for Labelled Classes 18m
Words and Strings 12m
Labelled trees 15m
Mappings 17m
Summary 4m
Exercises 2m
Exercises from Lecture 210m
1 practice exercise
Labeled Structures and EGFs4m
Week
3
2 hours to complete

## 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 videos (Total 84 min), 1 reading, 1 quiz
5 videos
Moment Calculations 24m
OBGF examples 17m
Labelled Classes 19m
Exercises 2m
Exercises from Lecture 310m
1 practice exercise
Combinatorial Parameters and MGFs8m
Week
4
2 hours to complete

## 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 videos (Total 109 min), 1 reading, 1 quiz
6 videos
Complex Functions 13m
Rational Functions 19m
Analytic Functions and Complex Integration 23m
Meromorphic Functions 34m
Exercises 3m
Exercises from Lecture 410m
1 practice exercise
Complex Analysis, Rational and Meromorphic Asymptotics4m

## Instructor

### Robert Sedgewick

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