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Yuchun Ma

Our lives are full of combinations. Combinatorial mathematics is just the science to deal with combinations of discrete items. As an ancient field, the history of combinatorial mathematics can be traced back over 4000 years to the age of the Great Yu in ancient China. Today, combinatorial mathematics is regarded as the basis of computer science since the algorithms in programming heavily rely on the analysis of the discrete elements.

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Our lives are full of combinations. Combinatorial mathematics is just the science to deal with combinations of discrete items. As an ancient field, the history of combinatorial mathematics can be traced back over 4000 years to the age of the Great Yu in ancient China. Today, combinatorial mathematics is regarded as the basis of computer science since the algorithms in programming heavily rely on the analysis of the discrete elements.

Instead of relying on the traditional mathematical "theorem - proof" format, this course demonstrates various principles in an intuitive manner with ancient stories, the scenes of movies and even a magic show. What you’ll learn:

  • The counting principles based on the basic operations “+”, “-”, “*”, “/”;
  • Generating functions
  • Recurrent number serials such as Fibonacci number, Catalan number, and more
  • Pigeon hole principles
  • Inclusion and exclusion principles
  • Polya counting based on group theory

This course is based on a highly regarded on-campus Tsinghua class called Combinatorics, and is ideal for students who are interested in mathematics or computer science. Enroll today and learn the mathematical theory needed to solve the real-world problems!

Every aspect of our lives is full of combinations, and combinatorics is the study of discrete things. As a subject with a long history, the development of combinatorics can be traced back to the era of Dayu 4,000 years ago. Nowadays, combinatorial mathematics has opened a new chapter with the development of computer science. Since the analysis and implementation of program algorithms are based on the analysis of discrete objects, combinatorial mathematics has developed into a basic theory of computer science.

This course changes the "theorem-proof" model of previous mathematics teaching and leads everyone to gradually explore the source of knowledge from the shallower to the deeper. There are ancient stories, movie clips, and even magic to interpret mathematical concepts. These forms are designed to lead everyone to feel the beauty of mathematics. Specific teaching content includes:

  • Counting rules based on four arithmetic operations;
  • parent function;
  • Recursive sequences, such as Fibonacci numbers, Cattelan numbers, etc.;
  • pigeonhole principle;
  • Inclusion-exclusion principle;
  • Polya's theorem based on group theory.

The content and outline of this course are mainly based on the excellent course "Combinatorics" of Tsinghua University. Through the study of this course, learners can have an in-depth understanding of the abstract theory and specific methods of counting, and thus deeply understand the role of combinatorial mathematics in promoting the development of computer theory. .

FAQ

I don’t speak Chinese, can I learn the course?

All the materials are in English. Though the original video was recorded in Chinese, the course team record the corresponding dubbing in English. All the audio and subtitles are processed to fit the English dubbing as much as possible, so that you can enjoy this wonderful course in English.

What are the textbook and the reference books for this course?

There is no textbook requirement for this course. The handouts distributed every week are critical. The following books are references

  • Richard A. Brualdi; Introductory Combinatorics (5th edition), Pearson, 2009
  • J.H.van Lint and R.M. Wilson; A course in Combinatorics, Cambridge University Press, 2001
  • Lu Kaicheng, "Combinatorics" fourth edition, Tsinghua University Press

What is the grading breakdown?

  • 70% quizzes and exercises
  • 30% final exam

How can I get the certificate?

If your final score is no less than 60.

Do I need to know how to program to learn this class?

Not necessary. This course is a math course which is based on fundamental theory. But to help the students to have the intuitive feel of the effects of the theory, we also provide a code lib that you can compare different implementations by running different programs.

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What's inside

Learning objectives

  • Counting principles in our daily lives
  • Applying math to computer science and financial analysis
  • The science behind combinations of discrete items
  • Understand the history of combinatorics in life and mathematics
  • In-depth study of abstract theories and concrete methods of counting
  • Deeply understand the role of combinatorial mathematics in promoting the development of computer theory

Good to know

Know what's good
, what to watch for
, and possible dealbreakers
Examines combinatorial mathematics, which is highly relevant to the analysis of discrete objects in computer science and data science
Develops counting principles and theory that help students understand how to efficiently solve counting problems in the real world
In-depth study of combinatorial mathematics, providing a strong foundation for further study in computer science, finance, and other fields that use discrete math
Taught by renowned professors from Tsinghua University, with a proven track record of excellence in combinatorial mathematics research and teaching
Explores the intriguing history of combinatorial mathematics in ancient China, making the subject more engaging and relatable
Provides a range of resources, including video lectures, handouts, and interactive exercises, to support student learning

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Reviews summary

Intuitive combinatorics

This intuitive course in Combinatorics, based on a university course by Tsinghua University, provides a deep understanding of counting.

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 Combinatorial Mathematics | 组合数学 with these activities:
Review 'Introductory Combinatorics' by Richard A. Brualdi
Familiarize yourself with the fundamental principles and techniques of combinatorics to provide a solid foundation for the course.
Show steps
  • Read Chapters 1 and 2 to gain an overview of counting principles and basic combinatorics.
  • Solve the practice problems at the end of each chapter to test your understanding. (Optional)
  • Attend the first few lectures to identify any areas where you may need additional support.
Attend a virtual study session with classmates
Engage with peers to discuss course concepts, clarify doubts, and reinforce your learning.
Show steps
  • Organize a virtual study session with 2-3 classmates.
  • Choose a specific topic or problem set to focus on.
  • Discuss the topic, ask questions, and work together to solve problems.
Solve practice problems from 'Combinatorics' by J.H. van Lint and R.M. Wilson
Reinforce your understanding of combinatorial techniques by solving a variety of practice problems.
Show steps
  • Choose a section from the book that aligns with the current course material.
  • Attempt to solve 5-10 problems from that section independently.
  • Review your solutions and identify any areas where you need further clarification.
Three other activities
Expand to see all activities and additional details
Show all six activities
Create a visual representation of a combinatorial problem
Enhance your problem-solving skills by visualizing combinatorial concepts.
Browse courses on Graph Theory
Show steps
  • Choose a combinatorial problem that interests you.
  • Represent the problem using a diagram, chart, or other visual format.
  • Analyze the visual representation to identify patterns and relationships.
Follow tutorials on Polya's Theorem and its applications
Enhance your understanding of Polya's Theorem and its use in counting problems.
Show steps
  • Search for online tutorials or videos that explain Polya's Theorem.
  • Follow along with the tutorial and work through the examples provided.
  • Apply the theorem to solve a few problems on your own.
Attend a workshop on combinatorial game theory
Expand your knowledge and explore real-world applications of combinatorial game theory.
Browse courses on Game Theory
Show steps
  • Identify and register for a workshop on combinatorial game theory.
  • Attend the workshop and engage actively in the discussions and exercises.
  • Reflect on the concepts learned and their potential applications in your own research or career.

Career center

Learners who complete Combinatorial Mathematics | 组合数学 will develop knowledge and skills that may be useful to these careers:
Professor
A Professor conducts research and teaches at a university. This course, Combinatorial Mathematics, can help build a foundation for this role by providing an understanding of the counting principles and methods used in mathematical research. The course also covers topics such as generating functions and recurrent number sequences, which are used in various mathematical disciplines.
Mathematician
A Mathematician develops and applies mathematical theories and methods. This course, Combinatorial Mathematics, can help build a foundation for this role by providing an understanding of the counting principles and methods used in mathematics. The course also covers topics such as generating functions and recurrent number sequences, which are used in various mathematical disciplines.
Data Scientist
A Data Scientist collects, analyzes, and interpret large amounts of complex data. This course, Combinatorial Mathematics, can help build a foundation for this role by providing an understanding of the counting principles and methods used in data analysis. The course also covers topics such as generating functions and recurrent number sequences, which are used in data modeling and forecasting.
Researcher
A Researcher conducts research in a particular field. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in research. The course also covers topics such as generating functions and recurrent number sequences, which are used in data analysis and modeling.
Data Analyst
A Data Analyst collects, analyzes, and interprets data. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in data analysis. The course also covers topics such as generating functions and recurrent number sequences, which are used in data modeling and forecasting.
Operations Research Analyst
An Operations Research Analyst develops and implements mathematical models to improve business operations. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in operations research. The course also covers topics such as generating functions and recurrent number sequences, which are used in logistics and supply chain management.
Quantitative Analyst
A Quantitative Analyst develops and implements mathematical models to analyze financial data. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in quantitative analysis. The course also covers topics such as generating functions and recurrent number sequences, which are used in financial modeling and forecasting.
Actuary
An Actuary evaluates and manages financial risks. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in actuarial science. The course also covers topics such as generating functions and recurrent number sequences, which are used in insurance and pension modeling.
Statistician
A Statistician collects, analyzes, and interprets data. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in statistics. The course also covers topics such as generating functions and recurrent number sequences, which are used in data modeling and forecasting.
Computer Scientist
A Computer Scientist designs, develops, and maintains computer systems. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in computer science. The course also covers topics such as generating functions and recurrent number sequences, which are used in software design and analysis.
Financial Analyst
A Financial Analyst evaluates investments and makes recommendations to clients. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in financial analysis. The course also covers topics such as generating functions and recurrent number sequences, which are used in financial modeling and forecasting.
Business Analyst
A Business Analyst analyzes business processes and makes recommendations for improvement. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in business analysis. The course also covers topics such as generating functions and recurrent number sequences, which are used in data modeling and forecasting.
Market Researcher
A Market Researcher conducts research to understand consumer behavior and trends. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in market research. The course also covers topics such as generating functions and recurrent number sequences, which are used in data analysis and forecasting.
Teacher
A Teacher educates students at various levels. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in teaching mathematics. The course also covers topics such as generating functions and recurrent number sequences, which are used in curriculum design and assessment.
Software Engineer
A Software Engineer designs, develops, and maintains software applications. This course, Combinatorial Mathematics, may be useful for this role as it provides an understanding of the counting principles and methods used in software engineering. The course also covers topics such as generating functions and recurrent number sequences, which are used in software design and analysis.

Reading list

We've selected eight 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 Combinatorial Mathematics | 组合数学.
Provides a comprehensive introduction to combinatorics, including topics such as counting, generating functions, and graph theory. It widely used textbook for undergraduate and graduate courses in combinatorics.
Classic textbook on combinatorics. It covers a wide range of topics, including counting, generating functions, and graph theory. It valuable reference for anyone interested in combinatorics.
这本教材全面系统地介绍了组合数学的基本理论及其在计算机科学、运筹学等领域的应用。
Is an excellent introduction to combinatorial mathematics and good supplement to the course. It provides a clear and concise overview of the subject and covers a wide range of topics.
Combines elementary graph theory and combinatorics in a single text. It discusses a number of topics of interest to undergraduates, including group actions on graphs, applications of combinatorics to graph coloring, and graph theory to the enumeration of chemical compounds.
Provides a collection of combinatorial problems and exercises, with solutions. It valuable resource for students and researchers in combinatorics.
This advanced textbook provides a comprehensive treatment of combinatorics, covering topics such as counting, generating functions, and graph theory. It valuable reference for anyone interested in the latest developments in combinatorics.

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