Combinatorics  

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

Branches > s.a. Coloring; Combinatorial Group Theory; Combinatorial Topology.
* Enumeration theory: Its purpose is to determine, given a system {Si of finite sets, the cardinality of each Si, or counting function N(i); Examples: For Sn = P{1,...,n}, N(n) = 2n; For Sn = {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; phase transitions.
@ Texts: Erdős & Spencer 74; Alon & Spencer 00; Beck 09 [inevitable randomness in discrete mathematics].
@ And physics: Scott & Sokal JSP(05)cm/03 [repulsive lattice gas].

Other Concepts > s.a. Generating Function; partitions; Species [combinatorial species].
* 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).
> And other areas: see grassmann numbers.

References > s.a. graphs.
@ General: Vilenkin 71; Comtet 74; Street & Wallis 77; Rota 78; Aigner 79; Stanley 83; Penner 99 [proof techniques]; Bóna 02 [II/II].
@ Books, I: Niven 65; Honsberger 73, 76 [problems].
@ Books, II: Cen & Koh 92 [problems]; Andreescu & Feng 02 [International Mathematical Olympiad problems], 03 [counting strategies]; Bóna 11 [enumeration and graph theory]; Vermani & Vermani 12 [discrete mathematics]; Koh & Tay 13 [counting].
@ 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 al qp/06 [integer sequences and quantum field theories]; Duchamp & Cheballah a0901 [open problems in combinatorial physics]; Tanasa SLC-a1102 [in quantum field theory].


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