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 {*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*)
= 2^{n};
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.

@ __References__:
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 and Results** > 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).

@ __References__:
Dorlas et al a1902 [identity
relating collections of *m* complex numbers and partitions of {1, ..., *m*}].

> __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];
West 20.

@ __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];
> s.a. quantum computing.

**In Physics** > s.a. Polymers;
probability in physics; states
in statistical mechanics [partition function]; tilings.

* __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.

@ __General references__: Duchamp & Cheballah a0901 [open problems in combinatorial physics].

@ __In quantum field theory__: Bender et al qp/06 [and integer sequences];
Tanasa SLC-a1102;
> s.a. quantum field theory formalism.

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send feedback and suggestions to bombelli at olemiss.edu – modified 15 jul 2020