Combinatorics

In General > s.a. discrete geometry.
* Idea: Combinatorial theory is the branch of mathematics concerned with finite problems of counting (how many elements there are in sets that are known to be finite), selection, arrangement, permutation, etc.
> Online resources: from MathWorld.

Branches > s.a. Coloring; Combinatorial Group Theory; Combinatorial Topology.
* Enumeration theory: Its purpose is to determine, given a system {S_i of finite sets, the cardinality of each S_i, or counting function N(i); Examples: For S_n = P{1,...,n}, N(n) = 2ⁿ; For S_n = {divisors of n}, N(n) = d(n).
* Other: It includes Ramsey theory, combinatorial designs, codes, graphs, networks, finite Boolean Algebras, game theory, finite probability theory, combinatorial geometry, lattices, Matroids, posets.

Algebraic Combinatorics
@ References: Stanley BAMS(03) [progress].

Combinatorial Geometry > s.a. Geometric Topology.
$ Def: A matroid in which all single points and pairs are independent sets.
@ Texts: Crapo & Rota 70; Pach & Agarwal 95.
@ Reference: Goodman & O'Rourke 04 [discrete methods in geometry].

Combinatorial Topology > see cell complexes.

Probabilistic Combinatorics > s.a. graphs.
@ Texts: Erdös & Spencer 74; Alon & Spencer 00.
@ And physics: Scott & Sokal JSP(05)cm/03 [repulsive lattice gas].

Other Concepts > s.a. partitions.
* Combinatorial numbers: The best known ones are binomial numbers; Other examples are Rook, Bell and Stirling numbers, which find applications in quantum field theory (normal ordering of operators).

References > s.a. graphs.
@ General: Comtet 74; Street & Wallis 77; Rota 78; Aigner 79; Stanley 83.
@ II: Andreescu & Feng 04.
@ Simple books: Niven 65.
@ Simple problems: Honsberger 73, 76.
@ Computational: Pemmaraju & Skiena 03 [Mathematica].

In Physics > s.a. Polymers; probability in physics; states in statistical mechanics [partition function]; tiling.
* Idea: Traditionally, among physicists combinatorics was identified with enumeration theory and probabilistic combinatorics, but lattice theory (from quantum mechanics) and graph and poset theory (from quantum gravity, for example) are becoming more important and better known.
@ References: Bender et all qp/06 [integer sequences and quantum field theories].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 21 jun 2008