Noncrossing partitions. (English) Zbl 0959.05009

Simion, Rodica

Discrete Math. 217, No. 1-3, 367-409 (2000).

A partition of \(\{1,\ldots,n\}\) is noncrossing if whenever \(1\leq a < b < c < d \leq n\) and \(a,c\) are in the same class and \(b,d\) are in the same class, then the two classes coincide. The enumerative study of noncrossing partitions was originated by G. Kreweras in his seminal 1972 paper; see G. Kreweras [Discrete Math. 1, 333-350 (1972; Zbl 0231.05014)]. Meanwhile it turned out that the lattice of noncrossing partitions has a wide interaction with algebraic, geometric combinatorics, probability measures, mathematical biology, etc. This long paper gives a detailed survey on theoretical results on noncrossing partitions as well as its applications on different topics.

Reviewer: Peter L.Erdős (Budapest)

MSC:

05A18	Partitions of sets
05A15	Exact enumeration problems, generating functions
05E10	Combinatorial aspects of representation theory
92B99	Mathematical biology in general
46L54	Free probability and free operator algebras
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
05A10	Factorials, binomial coefficients, combinatorial functions
05A30	\(q\)-calculus and related topics

Keywords:

noncrossing partition; algebraic combinatorics; topology