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.

@ __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]; 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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12 feb 2018