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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
Week
5
1 hour to complete

## Applications of Rational and Meromorphic Asymptotics

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....
6 videos (Total 63 min), 1 reading, 1 quiz
6 videos
Other Familiar Examples 15m
Restricted Compositions 9m
Supercritical Sequence Schema 14m
Summary 3m
Exercises 2m
Exercises from Lecture 510m
1 practice exercise
Applications of Complex Analysis, Rational and Meromorphic Asymptotics4m
Week
6
2 hours to complete

## Singularity Analysis

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....
5 videos (Total 82 min), 1 reading, 1 quiz
5 videos
Standard Function Scale 11m
Singularity Analysis 20m
Schemas and Transfer Theorems 25m
Exercises 2m
Exercises from Lecture 610m
1 practice exercise
Singularity Analysis of Generating Functions4m
Week
7
1 hour to complete

## Applications of Singularity Analysis

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....
6 videos (Total 65 min), 1 reading, 1 quiz
6 videos
Labelled Sets 15m
Mappings 11m
Tree-like Classes 16m
Summary 3m
Exercises 2m
Exercises from Lecture 710m
1 practice exercise
Applications of Singularity Analysis4m
Week
8
1 hour to complete

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....
5 videos (Total 63 min), 1 quiz
5 videos
Applications 5m
AC Wrap-up 11m
1 practice exercise

## Instructor

### Robert Sedgewick

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