--- title: Combinatorics description: Combinatorics is the study of finite or countable discrete structures. Learners will understand how to count arrangements, analyze permutations and combinations, use generating functions, and apply graph theory to solve complex optimization and partitioning problems. category: mathematics subcategory: discrete-math difficulty: beginner, intermediate, advanced url: /subject/combinatorics --- # Combinatorics Combinatorics is the study of finite or countable discrete structures. Learners will understand how to count arrangements, analyze permutations and combinations, use generating functions, and apply graph theory to solve complex optimization and partitioning problems. ## Where to start Start with Mathematics for Computer Science (MIT 6.042J) on MIT OpenCourseWare, a free course covering counting, graph theory and discrete probability, with lecture videos and problem sets with solutions. If you only use one resource, make it Richard P. Stanley's Enumerative Combinatorics, Volume 1, the authoritative text on counting. Before tackling it, Concrete Mathematics by Graham, Knuth and Patashnik builds fluency with binomial coefficients and generating functions. ## Available Resources 7 Books • 5 Courses • 2 Websites ## Courses ### 1. Probabilistic Methods in Combinatorics (MIT 18.226) **Author:** Yufei Zhao A graduate introduction to proving combinatorial objects exist by showing random constructions work with positive probability: linearity of expectation, alterations, second moment method, Lovász local lemma, and concentration inequalities. 11 lecture videos, lecture notes, Zhao's open textbook and problem sets. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-226-probabilistic-methods-in-combinatorics-fall-2022/ **Tags:** probabilistic-method, lovasz-local-lemma, second-moment-method, concentration-inequalities, random-graphs ### 2. Graph Theory and Additive Combinatorics (MIT 18.225) **Author:** Yufei Zhao Classical and modern results linking graph theory and additive combinatorics: Turán-type extremal problems, Szemerédi's regularity lemma, pseudorandom graphs, graph limits, Roth's theorem and Freiman's theorem, plus open problems. 26 lecture videos, lecture notes, Zhao's open textbook and problem sets. **Difficulty:** Advanced | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-225-graph-theory-and-additive-combinatorics-fall-2023/ **Tags:** extremal-graph-theory, additive-combinatorics, regularity-lemma, pseudorandom-graphs, graph-limits ### 3. Graph Theory and Additive Combinatorics Explore graph theory and additive combinatorics! This course covers key concepts and applications in graphs & digraphs. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX **Tags:** graph-theory, additive-combinatorics, extremal-combinatorics, szemeredi-regularity, lecture-series ### 4. Mathematics for Computer Science (MIT 6.042J) **Author:** Tom Leighton, Marten van Dijk Discrete mathematics for computer science with an emphasis on definitions and proofs: logic, induction, sets and relations, graph theory, modular arithmetic, asymptotics, counting and discrete probability. 25 lecture videos, problem sets and exams with solutions build fluency in writing proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/ **Tags:** discrete-mathematics, mathematical-proofs, induction, graph-theory, combinatorics, discrete-probability ### 5. Discrete Mathematics **Author:** Dominik Scheder Dominik Scheder's proof-based course on sets, functions and relations, enumerative combinatorics, graph theory, and network flows and matchings, each concept paired with a fully proved non-trivial result. Learners read formal statements and write rigorous proofs of their own. **Difficulty:** Intermediate | **Language:** English | **Duration:** 11 weeks of study, 3-5 hours per week. | **Price:** Free **Link:** https://www.coursera.org/learn/discrete-mathematics **Tags:** discrete-math, combinatorics, graph-theory, network-flows, proofs ## Websites ### 1. Art of Problem Solving - Combinatorics The Art of Problem Solving wiki's combinatorics hub, linking concise articles on counting principles, permutations, combinations, the binomial theorem, stars and bars, and inclusion-exclusion, with competition problems. Suited to students preparing for contests such as AMC and AIME. **Difficulty:** Beginner | **Price:** Free **Link:** https://artofproblemsolving.com/wiki/index.php/Combinatorics **Tags:** combinatorics, combinations, counting, competition-math, binomial-theorem ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Books ### 1. Concrete Mathematics - Graham, Knuth, Patashnik **Author:** Ronald L. Graham, Donald E. Knuth, Oren Patashnik Covers sums, recurrences, integer functions, binomial coefficients, generating functions, and asymptotics with the notation used in algorithm analysis. After working its exercises you can manipulate summations and solve recurrences in closed form. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://amzn.to/49WVhIs **Tags:** discrete-math, generating-functions, recurrences, asymptotics, algorithm-analysis ### 2. Discrete Mathematics and its Applications - Kenneth Rosen **Author:** Kenneth H. Rosen Standard undergraduate textbook covering logic, proof techniques, set theory, counting, recurrences, and graphs, with a number theory chapter on divisibility, modular arithmetic, primes and RSA. Extensive exercise sets follow each section for self-study practice. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://amzn.to/4k8ABBR **Tags:** discrete-mathematics, proof-techniques, combinatorics, modular-arithmetic, textbook ### 3. Introduction to Combinatorics **Author:** Martin J. Erickson Wiley text organized around existence, enumeration, and construction, building from first principles to van der Waerden's theorem, Polya's enumeration formula, and the Leech lattice. Suited to advanced undergraduates comfortable with proof-based mathematics. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0471154083?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, discrete-math ### 4. Enumerative Combinatorics, Volume 1 **Author:** Richard P. Stanley Graduate-level reference on counting: sieve methods, partially ordered sets and Möbius inversion, rational generating functions, and permutation statistics such as descents and inversions. Its exercises, each rated by difficulty and with solutions, train readers to find bijective and generating-function proofs. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521663512?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, combinatorics ### 5. Enumerative Combinatorics, Volume 2 **Author:** Richard P. Stanley Graduate text on composition of generating functions, trees, algebraic and D-finite generating functions, and symmetric functions, including Schur functions and the RSK algorithm, with an appendix by Sergey Fomin and hundreds of exercises. Readers learn advanced counting techniques used in algebraic combinatorics research. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521169879?tag=edmonddante07-20 **Tags:** combinatorics, enumerative-combinatorics, generating-functions, symmetric-functions, algebraic-combinatorics ### 6. Introduction to Combinatorial Analysis **Author:** John Riordan Classic 1958 monograph, reprinted by Dover, on generating-function methods for enumeration: permutations and combinations, the principle of inclusion and exclusion, permutations with restricted positions, cycle structure and distributions. Shows how to set up and solve counting problems via recurrences and generating functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486425363?tag=edmonddante07-20 **Tags:** combinatorics, enumerative-combinatorics, generating-functions, inclusion-exclusion, recurrence-relations ### 7. Probabilistic Combinatorics and Its Applications **Author:** Fan R. K. Chung Lecture notes from a 1991 AMS short course edited by Béla Bollobás. Surveys of random graphs, martingale and isoperimetric inequalities, eigenvalue methods, branching processes and rapidly mixing Markov chains show how probabilistic arguments settle combinatorial problems. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/082185500X?tag=edmonddante07-20 **Tags:** probabilistic-combinatorics, random-graphs, probabilistic-method, markov-chains, martingale-inequalities --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/combinatorics