Best resources to learnCombinatorics

Probabilistic Methods in Combinatorics (MIT 18.226)

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.

Graph Theory and Additive Combinatorics (MIT 18.225)

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.

Graph Theory and Additive Combinatorics

Explore graph theory and additive combinatorics! This course covers key concepts and applications in graphs & digraphs.

Mathematics for Computer Science (MIT 6.042J)

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.

Discrete Mathematics

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.

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Concrete Mathematics - Graham, Knuth, 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.

Discrete Mathematics and its Applications - Kenneth 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.

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Introduction to Combinatorics

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.

Enumerative Combinatorics, Volume 1

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.

Enumerative Combinatorics, Volume 2

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.

Introduction to Combinatorial Analysis

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.

🌐Websites(2)

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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.

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Wolfram MathWorld

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.