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Analytic Combinatorics

Robert Sedgewick

4.7

Based on 67 ratings

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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.

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.

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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.

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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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Reviews summary

Rigorous analytic combinatorics: theory and methods

According to learners, Analytic Combinatorics is a highly theoretical and rigorous course from Princeton University that offers excellent depth into the subject. Students report it is a challenging course, often requiring a strong mathematical background, particularly in areas like discrete math and possibly real/complex analysis fundamentals before starting. The course focuses heavily on generating functions and applying complex analysis techniques to solve combinatorial problems. Many feel it provides a strong foundation for advanced study or research in the field, though its highly academic nature means it's less focused on practical, immediate applications.

Focuses on theory, less on application.

"This is a very theoretical course, which is great if you're into academics."

"Don't expect many direct coding or applied examples; it's pure mathematics."

"It's best suited for those interested in the fundamental theory of combinatorics."

Prerequisites in math/CS are essential.

"Make sure you have a solid background in discrete math and calculus before taking this."

"Prior exposure to complex numbers or analysis would be very helpful to keep pace."

"I felt a bit lost at times without stronger prerequisite knowledge."

Integrates complex analysis techniques.

"The way complex analysis is applied to derive asymptotics is fascinating and powerful."

"The introduction to complex analysis within the course was clear enough to follow the applications."

"I appreciated learning how analytic methods using complex functions yield asymptotic results."

Provides thorough coverage of theory.

"The course goes into excellent depth on generating functions and asymptotics."

"I really appreciate the comprehensive coverage of both symbolic and analytic methods."

"It gave me a deep understanding of the theoretical underpinnings."

Builds base for advanced study/research.

"This course is essential for anyone doing research in combinatorics or algorithms."

"It gave me the precise tools I needed for my own research work."

"I found this course highly valuable for my graduate studies."

Rigorous content requires dedication.

"This is definitely a challenging course; be prepared for complex math."

"It requires significant effort to keep up, especially the complex analysis parts."

"I found the material quite rigorous; it's not for the faint of heart or beginners."

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

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Gather and organize resources related to combinatorial structures, providing a comprehensive reference for further exploration.

Browse courses on Combinatorics

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Identify and collect relevant articles, videos, and online resources.
Organize the resources into a structured and accessible format.

Review Limits and Derivatives

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Revisit foundational math concepts to strengthen the understanding of topics like generating functions and asymptotics.

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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

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Reinforce the understanding of constructing generating function equations for various combinatorial constructions.

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Solve practice problems on deriving generating functions for combinatorial structures.
Analyze examples of generating function equations and identify the corresponding combinatorial constructions.

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Collaboration and Discussion Sessions

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Engage with peers to discuss course concepts, solve problems together, and enhance understanding through diverse perspectives.

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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

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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

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Read the relevant chapters of the book.
Work through the exercises and examples provided in the book.

Tutoring in Combinatorics

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Reinforce understanding by sharing knowledge with others, clarifying concepts, and solidifying learning through teaching.

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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

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Apply analytical methods to derive asymptotic estimates for combinatorial structures, enhancing comprehension of complex analysis concepts.

Browse courses on Analytic Combinatorics

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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

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Explore advanced asymptotic techniques beyond the course material, deepening the understanding of asymptotic estimates.

Browse courses on Asymptotics

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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 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.

See salaries and explore the career path for Operations Research 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.

See salaries and explore the career path for Financial Analyst

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.

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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.

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Computer Scientist

Computer Scientists analyze, design, implement, and manage the software and hardware systems used to solve problems.

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Statistician

Statisticians collect, analyze, and interpret data to help organizations and individuals make informed decisions.

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Business Analyst

Business Analysts evaluate business processes and systems to improve efficiency and effectiveness.

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Actuary

Actuaries use mathematical and statistical methods to assess the probability of uncertain future events.

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Quantitative Analyst

Quantitative Analysts use mathematical and statistical models to assess the risk and return of financial investments.

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Risk Manager

Risk Managers identify, assess, and manage risks to an organization's operations and finances.

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Consultant

Consultants provide expert advice to organizations on a wide range of topics, including business strategy, operations, and technology.

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Mathematician

Mathematicians study the properties of numbers, structures, and change.

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Physicist

Physicists study the fundamental laws that govern the universe.

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Chemist

Chemists study the properties and behavior of matter.

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Biologist

Biologists study the structure, function, growth, and evolution of living organisms.

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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.

An Introduction to the Analysis of Algorithms (2nd...

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The textbook for this course. Goes over all of the material, and provides lots of practice problems and exercises.

Analytic Combinatorics

What's inside

Syllabus

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Reviews summary

Rigorous analytic combinatorics: theory and methods

Activities

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Reading list

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