Combinatorics
Combinatorics studies finite and discrete configurations through counting, extremal structure, and constructive arguments.
Additive combinatorics is the largest field, with about one article in seven. It runs from zero-sum problems in finite abelian groups and sumsets and inverse theorems, from the Cauchy–Davenport–Chowla theorem to the polynomial Freiman–Ruzsa theorem over \(\mathbb{F}_2\) of Gowers, Green, Manners and Tao, to Sidon sets and difference sets and progressions in dense sets, with Szemerédi's theorem. Ramsey theory has the theorems of Ramsey, van der Waerden and Schur, the density and infinitary Hales–Jewett theorems, and Hindman's finite sums theorem.
Combinatorics on words, nearly as large, studies morphic and Sturmian words such as the Fibonacci and Thue–Morse words, automatic sequences with the theorems of Cobham and Christol, and finite words, where Witt's formula counts Lyndon words of given content. Permutations are studied through pattern avoidance and statistics, and Boolean functions include the Kahn–Kalai–Linial inequality: some variable of a Boolean function on \(n\) bits has influence at least of order \(\operatorname{Var}(f) \log n / n\).
Eight enumerative fields hold over a third of the articles between them: Stirling, Bell and Eulerian numbers, integer partitions, binomial coefficients and their sums, Catalan numbers and lattice paths, integer compositions, Riordan arrays, parking functions and set partitions.
- Binomial coefficients and identities
- Catalan numbers and lattice paths
- Stirling, Bell and Eulerian numbers
- Riordan arrays and sequence transforms
- Integer partitions
- Integer compositions
- Permutations
- Set partitions
- Parking functions
- Tilings, dimers and board placements
- Symmetric functions, tableaux and q-analogues
- Combinatorics on words
- Additive combinatorics
- Ramsey theory
- Extremal set theory and hypergraphs
- Designs, Latin squares and combinatorial matrices
- Matroid theory
- Partially ordered sets
- Polytopes, Ehrhart theory and simplicial complexes
- Discrepancy theory
- Probabilistic combinatorics
- Boolean functions
- Elementary enumeration
Loading dependency explorer…
The dependency explorer needs JavaScript and its graph data.