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Noncrossing partitions. (English) Zbl 0959.05009
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.

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
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