Analytic Combinatorics

Princeton University via Coursera

Go to Course: https://www.coursera.org/learn/analytic-combinatorics

Introduction

### Course Review: Analytic Combinatorics on Coursera Are you a mathematics enthusiast or a computer science student looking to delve deeper into the world of combinatorial structures? If so, the **Analytic Combinatorics** course offered on Coursera might just be the right fit for you. This course presents an intricate look at analytic methods used for making precise quantitative predictions about large combinatorial structures. #### Course Overview **Analytic Combinatorics** teaches a unique calculus specifically designed to derive functional relationships among various generating functions: ordinary (OGFs), exponential (EGFs), and multivariate generating functions (MGFs). The course also integrates methods from complex analysis to derive accurate asymptotic behaviors from these generating function equations. The best part? All features of this course are available for free, although it does not provide a certificate upon completion, making it accessible to anyone eager to learn without the pressure of certification. #### Syllabus Breakdown The course is structured into systematically organized modules that gradually build your understanding of analytic combinatorics: 1. **Combinatorial Structures and OGFs**: - This initial lecture introduces the symbolic method of combinatorial constructions. You will learn how to define classes of combinatorial objects and derive generating functions. - Expect a variety of classical examples that solidify these concepts. 2. **Labelled Structures and EGFs**: - Here, you will work with labelled objects, integrating exponential generating functions to study combinatorial classes. This module emphasizes understanding the construction of EGF equations, reinforcing your foundational knowledge with numerous examples. 3. **Combinatorial Parameters and MGFs**: - This lecture discusses marking parameters using variables, leading to multivariate generating functions. By exploring bivariate generating functions, you will learn to compute important statistical moments, heightening your grasp of combinatorial analysis. 4. **Complex Analysis, Rational and Meromorphic Asymptotics**: - This module introduces complex analysis, treating generating functions as analytic objects. Even if you have no prior knowledge of complex analysis, the course starts from the basics, ensuring you understand the necessary concepts to grasp asymptotic coefficient estimation. 5. **Applications of Rational and Meromorphic Asymptotics**: - You will explore classic combinatorial classes through the lens of the general transfer theorem, supplemented by universal laws governing a wide range of combinatorial classes. 6. **Singularity Analysis**: - An essential part of analytic combinatorics, this lecture covers the Flajolet-Odlyzko theorem. You will learn how to analyze generating functions to derive asymptotic behaviors systematically. 7. **Applications of Singularity Analysis**: - Once you've grasped the basics, this section applies the Flajolet-Odlyzko approach to classic combinatorial problems, reinforcing your understanding with practical applications. 8. **Saddle Point Asymptotics**: - Finally, you will explore the saddle point method, a powerful technique for deriving coefficient asymptotics for generating functions with no singularities, applying this approach to several classical problems. #### Recommendation **Analytic Combinatorics** is a course highly recommended for anyone keen on expanding their mathematical toolkit in combinatorial analysis. It’s perfect for graduate students, researchers, or anyone with a strong mathematical background. The course strikes a balance between theory and application, making it an excellent investment of time for serious learners. Moreover, since the course content is entirely free of charge and accessible on Coursera, there’s no reason not to explore it! While it may not grant you an official certificate upon completion, the skills and knowledge you acquire will undoubtedly be valuable in fields requiring combinatorial reasoning and analysis, such as computer science, statistics, or operations research. In conclusion, if you are ready to tackle the complexities of analytic combinatorics and wish to enhance your understanding of generating functions and asymptotic analysis, enroll in this course on Coursera and embark on a fascinating mathematical journey!

Syllabus

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.

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.

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.

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.

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.

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.

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.

Saddle Point Asymptotics

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.