Concrete Mathematics - Graham, Knuth, Patashnik
by 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.
More resources on Combinatorics
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.
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.
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.
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.