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Robert Sedgewick
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.
Read more
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.
Read more
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. People who are interested in digging deeper into the content may wish to obtain the textbook Analytic Combinatorics (upon which the course is based) or to visit the website ac.cs.princeton.edu for a wealth of additional material.
This course does not offer a certificate upon completion.
Register for this course and see more details by visiting:
OpenCourser.com/course/uxv62q/analytic
What's inside
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.
Read more
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.
Read more
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.
Good to know
Good to know
Know what's good ,
what to watch for , and
possible dealbreakers
Introduces learners to advanced quantitative tools such as generating functions and complex analysis
Applicable to various fields including computer science, operations research, and physics
Provides a strong foundation for those interested in pursuing research in combinatorial structures
Taught by Robert Sedgewick, a renowned computer scientist and author
No certificate of completion is offered
Save this course
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Reviews summary
Beautiful but advanced combinatorics
Learners say that Analytic Combinatorics is a difficult course that's well-received by students who appreciate its well-organized lectures. The instructor is knowledgeable, but the exams are difficult.
Learners can retake quizzes to test understanding.
"I appreciated the option to retake quizzes..."
Well-organized and engaging.
"The lectures are very well organized."
"Pr Sedgewick really knows what he talks about."
Very knowledgeable subject matter expert.
"The instructor, a most prominent figure..."
"Pr Sedgewick really knows what he talks about."
May be too challenging for some learners.
"This course is very hard..."
"The exams are difficult."
Activities
Be better prepared
before
your course. Deepen your understanding
during
and
after
it. Supplement your coursework and achieve mastery of the topics covered
in Analytic Combinatorics with these
activities:
Combinatorial Structures Resource Collection
Show steps
Gather and organize resources related to combinatorial structures, providing a comprehensive reference for further exploration.
Browse courses on
Combinatorics
Show steps
-
Identify and collect relevant articles, videos, and online resources.
-
Organize the resources into a structured and accessible format.
Review Limits and Derivatives
Show steps
Revisit foundational math concepts to strengthen the understanding of topics like generating functions and asymptotics.
Browse courses on
Limits
Show steps
-
Review the definition and properties of limits.
-
Practice finding limits of functions.
-
Review the definition and rules of derivatives.
-
Practice finding derivatives of functions.
Generating Function Equations Practice
Show steps
Reinforce the understanding of constructing generating function equations for various combinatorial constructions.
Browse courses on
Generating Functions
Show steps
-
Solve practice problems on deriving generating functions for combinatorial structures.
-
Analyze examples of generating function equations and identify the corresponding combinatorial constructions.
Five other activities
Expand to see all activities and additional details
Show all eight activities
Collaboration and Discussion Sessions
Show steps
Engage with peers to discuss course concepts, solve problems together, and enhance understanding through diverse perspectives.
Browse courses on
Collaboration
Show steps
-
Join or form a study group with other students.
-
Meet regularly to discuss course material, ask questions, and work on problems.
Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick
Show steps
Delve deeper into the concepts covered in the course by reading the foundational textbook, enhancing comprehension and reinforcing learning.
View
An Introduction to the Analysis of Algorithms
on Amazon
Show steps
-
Read the relevant chapters of the book.
-
Work through the exercises and examples provided in the book.
Tutoring in Combinatorics
Show steps
Reinforce understanding by sharing knowledge with others, clarifying concepts, and solidifying learning through teaching.
Browse courses on
Tutoring
Show steps
-
Identify tutoring opportunities at local schools or community centers.
-
Prepare lesson plans and materials for tutoring sessions.
-
Tutor students in combinatorial concepts and techniques.
Mathematical Analysis Project
Show steps
Apply analytical methods to derive asymptotic estimates for combinatorial structures, enhancing comprehension of complex analysis concepts.
Browse courses on
Analytic Combinatorics
Show steps
-
Choose a combinatorial structure of interest.
-
Define the generating function for the structure.
-
Apply complex analysis techniques to derive asymptotic estimates.
-
Write a report summarizing the analysis and findings.
Advanced Asymptotic Techniques Tutorial
Show steps
Explore advanced asymptotic techniques beyond the course material, deepening the understanding of asymptotic estimates.
Browse courses on
Asymptotics
Show steps
-
Find and review tutorials on advanced asymptotic methods.
-
Practice applying these techniques to solve combinatorial problems.
Combinatorial Structures Resource Collection
Show steps
Gather and organize resources related to combinatorial structures, providing a comprehensive reference for further exploration.
Browse courses on
Combinatorics
Show steps
Show steps
Show steps
- Identify and collect relevant articles, videos, and online resources.
- Organize the resources into a structured and accessible format.
Review Limits and Derivatives
Show steps
Revisit foundational math concepts to strengthen the understanding of topics like generating functions and asymptotics.
Browse courses on
Limits
Show steps
Show steps
Show steps
- Review the definition and properties of limits.
- Practice finding limits of functions.
- Review the definition and rules of derivatives.
- Practice finding derivatives of functions.
Generating Function Equations Practice
Show steps
Reinforce the understanding of constructing generating function equations for various combinatorial constructions.
Browse courses on
Generating Functions
Show steps
Show steps
Show steps
- Solve practice problems on deriving generating functions for combinatorial structures.
- Analyze examples of generating function equations and identify the corresponding combinatorial constructions.
Five other activities
Expand to see all activities and additional details
Show all eight activities
Collaboration and Discussion Sessions
Show steps
Engage with peers to discuss course concepts, solve problems together, and enhance understanding through diverse perspectives.
Browse courses on
Collaboration
Show steps
Show steps
Show steps
- Join or form a study group with other students.
- Meet regularly to discuss course material, ask questions, and work on problems.
Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick
Show steps
Delve deeper into the concepts covered in the course by reading the foundational textbook, enhancing comprehension and reinforcing learning.
View
An Introduction to the Analysis of Algorithms
on Amazon
Show steps
Show steps
Show steps
- Read the relevant chapters of the book.
- Work through the exercises and examples provided in the book.
Tutoring in Combinatorics
Show steps
Reinforce understanding by sharing knowledge with others, clarifying concepts, and solidifying learning through teaching.
Browse courses on
Tutoring
Show steps
Show steps
Show steps
- Identify tutoring opportunities at local schools or community centers.
- Prepare lesson plans and materials for tutoring sessions.
- Tutor students in combinatorial concepts and techniques.
Mathematical Analysis Project
Show steps
Apply analytical methods to derive asymptotic estimates for combinatorial structures, enhancing comprehension of complex analysis concepts.
Browse courses on
Analytic Combinatorics
Show steps
Show steps
Show steps
- Choose a combinatorial structure of interest.
- Define the generating function for the structure.
- Apply complex analysis techniques to derive asymptotic estimates.
- Write a report summarizing the analysis and findings.
Advanced Asymptotic Techniques Tutorial
Show steps
Explore advanced asymptotic techniques beyond the course material, deepening the understanding of asymptotic estimates.
Browse courses on
Asymptotics
Show steps
Show steps
Show steps
- Find and review tutorials on advanced asymptotic methods.
- Practice applying these techniques to solve combinatorial problems.
Career center
Learners who complete Analytic Combinatorics will develop knowledge and skills
that may be useful to these careers:
Operations Research Analyst
Operations Research Analyst
Operations Research Analysts improve efficiency across the life of a product, process, or organization. Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. It introduces the symbolic method to derive functional relations among generating functions and methods in complex analysis for deriving accurate asymptotics from the generating function equations. This may be useful for an Operations Research Analyst who needs to understand how to analyze large combinatorial structures.
Financial Analyst
Financial Analyst
Financial Analysts study historic and current financial data to make predictions. They use these predictions to inform investment decisions for organizations and individuals. Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. It introduces the symbolic method to derive functional relations among generating functions and methods in complex analysis for deriving accurate asymptotics from the generating function equations. This may be useful for a Financial Analyst who needs to understand how to analyze historical data and make predictions.
Data Scientist
Data Scientist
Data Scientists gain valuable insights from data to inform decision making. Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. It introduces the symbolic method to derive functional relations among generating functions and methods in complex analysis for deriving accurate asymptotics from the generating function equations. This may be useful for a Data Scientist who needs to understand how to analyze data and make predictions.
Software Engineer
Software Engineer
Software Engineers design, develop, and maintain software systems. Analytic Combinatorics teaches a calculus that enables precise quantitative predictions of large combinatorial structures. It introduces the symbolic method to derive functional relations among generating functions and methods in complex analysis for deriving accurate asymptotics from the generating function equations. This may be useful for a Software Engineer who needs to understand how to analyze and design software systems.
Computer Scientist
Computer Scientist
Computer Scientists analyze, design, implement, and manage the software and hardware systems used to solve problems.
Statistician
Statistician
Statisticians collect, analyze, and interpret data to help organizations and individuals make informed decisions.
Risk Manager
Risk Manager
Risk Managers identify, assess, and manage risks to an organization's operations and finances.
Physicist
Physicist
Physicists study the fundamental laws that govern the universe.
Biologist
Biologist
Biologists study the structure, function, growth, and evolution of living organisms.
Chemist
Chemist
Chemists study the properties and behavior of matter.
Business Analyst
Business Analyst
Business Analysts evaluate business processes and systems to improve efficiency and effectiveness.
Consultant
Consultant
Consultants provide expert advice to organizations on a wide range of topics, including business strategy, operations, and technology.
Quantitative Analyst
Quantitative Analyst
Quantitative Analysts use mathematical and statistical models to assess the risk and return of financial investments.
Actuary
Actuary
Actuaries use mathematical and statistical methods to assess the probability of uncertain future events.
Mathematician
Mathematician
Mathematicians study the properties of numbers, structures, and change.
For more career information including salaries, visit:
OpenCourser.com/course/uxv62q/analytic
Reading list
We've selected 48 books
that we think will supplement your
learning. Use these to
develop background knowledge, enrich your coursework, and gain a
deeper understanding of the topics covered in
Analytic Combinatorics.
The textbook for this course. Goes over all of the material, and provides lots of practice problems and exercises.
This is the textbook upon which the course is based. Students who want more depth in the material should consider this text.
A two-volume work that provides a comprehensive treatment of enumerative combinatorics, including many topics covered in this course.
Another excellent reference for anyone interested in the combinatorics and generating functions that this course is based on.
This advanced textbook covers a wide range of topics in enumerative combinatorics. It good reference to have.
Provides a comprehensive treatment of combinatorial enumeration, including many of the topics covered in the course. It good resource for anyone who wants to learn more about this field.
A multi-volume work that provides a comprehensive treatment of computer science, including many topics that are relevant to this course.
A comprehensive and advanced textbook that covers many of the topics in the course in more detail.
This classic textbook provides background in the concepts and techniques used in combinatorial analysis.
Provides a strong background on asymptotics, which are an important topic in Analytic Combinatorics.
Save
Provides a strong background on complex analysis, which is used extensively in Analytic Combinatorics.
This classic textbook provides background in asymptotic analysis and complex functions.
Save
Provides background on much of the combinatorial theory used in Analytic Combinatorics, which would be helpful for getting a solid foundation in the basics of the math.
Save
Provides background in complex analysis that is useful for understanding the asymptotic analysis in the course.
A good resource on the basics of discrete mathematics for anyone who is new to it.
A good resource on the basics of combinatorial optimization for anyone who is new to it.
This classic textbook provides background in number theory, which is useful for understanding some of the analytic methods used in the course.
A classic text on analytic combinatorics that may be of interest to those wanting more depth.
A good resource on the basics of algorithms for anyone who is new to it.
Save
This classic textbook provides background in probability theory, which is useful for understanding some of the concepts used in the course.
This classic textbook provides background in statistical inference, which is useful for understanding some of the concepts used in the course.
Save
Provides a good introduction to complex variables, which are used in the analysis of generating functions. It is helpful for understanding the complex analysis of generating functions.
This textbook provides a practical introduction to combinatorics, which is useful for understanding some of the basic concepts used in the course.
This textbook provides background in data structures and algorithms, which is useful for understanding some of the computational methods used in the course.
This classic textbook provides background in algorithms, which is useful for understanding some of the computational methods used in the course.
Collection of papers on asymptotic methods in combinatorics, graphs, and analysis. It valuable resource for anyone who wants to learn more about these topics.
An introduction to analytic combinatorics with a focus on applications in computer science.
A textbook on generating functions with a focus on applications in number theory and combinatorics.
This textbook provides background in combinatorics and graph theory, which is useful for understanding some of the basic concepts used in the course.
Save
Provides a good introduction to many of the mathematical topics that are used in combinatorics, including generating functions, complex analysis, and differential equations. It good resource for anyone who wants to learn more about these topics.
This textbook provides background in number theory, which is useful for understanding some of the analytic methods used in the course.
Save
Classic text on analytic number theory. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about analytic number theory.
This advanced textbook provides background in analytic methods in probability theory, which is useful for understanding some of the advanced concepts used in the course.
Comprehensive treatment of advanced combinatorics. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about advanced combinatorics.
Provides a good introduction to combinatorics and graph theory, which are related to the topics covered in the course. It good resource for anyone who wants to learn more about these topics.
This advanced textbook provides background in asymptotic methods in combinatorics, which is useful for understanding some of the advanced concepts used in the course.
Comprehensive treatment of algebraic combinatorics. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about algebraic combinatorics.
Provides a good introduction to probability, which is used in the analysis of combinatorial structures. It is helpful for understanding the probabilistic analysis of generating functions.
Provides a good introduction to algorithms, which are used in the analysis of combinatorial structures. It is helpful for understanding the algorithmic analysis of generating functions.
Provides a good introduction to linear algebra, which is used in the analysis of combinatorial structures. It is helpful for understanding the linear algebra of generating functions.
Save
Provides a good introduction to calculus, which is used in the analysis of generating functions. It is helpful for understanding the calculus of generating functions.
Save
Comprehensive treatment of polygonal dissections. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about polygonal dissections.
Provides a good introduction to number theory, which is used in the analysis of combinatorial structures. It is helpful for understanding the number theory of generating functions.
Comprehensive treatment of number theory. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about number theory.
Save
Comprehensive treatment of advanced topics in number theory. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about advanced topics in number theory.
Comprehensive treatment of applied combinatorics. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about applied combinatorics.
Comprehensive treatment of data structures and algorithms. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about data structures and algorithms.
Comprehensive treatment of combinatorial algorithms. It provides a thorough treatment of the subject, and it valuable resource for anyone who wants to learn more about combinatorial algorithms.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/uxv62q/analytic
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Princeton University
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Robert Sedgewick
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Good to know
Introduces learners to advanced quantitative tools such as generating functions and complex analysis
Applicable to various fields including computer science, operations research, and physics
Provides a strong foundation for those interested in pursuing research in combinatorial structures
Taught by Robert Sedgewick, a renowned computer scientist and author
No certificate of completion is offered
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