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In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossing partitions of a set of ''n'' elements is the ''n''th
Catalan number In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles C ...
. The number of noncrossing partitions of an ''n''-element set with ''k'' blocks is found in the Narayana number triangle.


Definition

A
partition of a set In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every pa ...
''S'' is a set of non-empty, pairwise disjoint subsets of ''S'', called "parts" or "blocks", whose union is all of ''S''. Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclic order like the vertices of a regular ''n''-gon. No generality is lost by taking this set to be ''S'' = . A noncrossing partition of ''S'' is a partition in which no two blocks "cross" each other, i.e., if ''a'' and ''b'' belong to one block and ''x'' and ''y'' to another, they are not arranged in the order ''a x b y''. If one draws an arch based at ''a'' and ''b'', and another arch based at ''x'' and ''y'', then the two arches cross each other if the order is ''a x b y'' but not if it is ''a x y b'' or ''a b x y''. In the latter two orders the partition is noncrossing. Equivalently, if we label the vertices of a regular ''n''-gon with the numbers 1 through ''n'', the convex hulls of different blocks of the partition are disjoint from each other, i.e., they also do not "cross" each other. The set of all non-crossing partitions of ''S'' is denoted \text(S). There is an obvious order isomorphism between \text(S_1) and \text(S_2) for two finite sets S_1,S_2 with the same size. That is, \text(S) depends essentially only on the size of S and we denote by \text(n) the non-crossing partitions on ''any'' set of size ''n''.


Lattice structure

Like the set of all partitions of the set , the set of all noncrossing partitions is a lattice when partially ordered by saying that a finer partition is "less than" a coarser partition. However, although it is a subset of the lattice of all partitions, it is ''not'' a sublattice of the lattice of all partitions, because the join operations do not agree. In other words, the finest partition that is coarser than both of two noncrossing partitions is not always the finest ''noncrossing'' partition that is coarser than both of them. Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions of a set is self-dual, i.e., it is order-isomorphic to the lattice that results from inverting the partial order ("turning it upside-down"). This can be seen by observing that each noncrossing partition has a complement. Indeed, every interval within this lattice is self-dual.


Role in free probability theory

The lattice of noncrossing partitions plays the same role in defining free cumulants in free probability theory that is played by the lattice of ''all'' partitions in defining joint cumulants in classical
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
. To be more precise, let (\mathcal,\phi) be a non-commutative probability space (See free probability for terminology.), a\in\mathcal a non-commutative random variable with free cumulants (k_n)_. Then :\phi(a^n) = \sum_ \prod_ k_j^{N_j(\pi)} where N_j(\pi) denotes the number of blocks of length j in the non-crossing partition \pi. That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. This is the free analogue of the moment-cumulant formula in classical probability. See also
Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...
.


References

*Germain Kreweras, "Sur les partitions non croisées d'un cycle", ''
Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continu ...
'', volume 1, number 4, pages 333–350, 1972. * Rodica Simion, "Noncrossing partitions", ''Discrete Mathematics'', volume 217, numbers 1–3, pages 367–409, April 2000.
Roland Speicher, "Free probability and noncrossing partitions"

Séminaire Lotharingien de Combinatoire
', B39c (1997), 38 pages, 1997 Families of sets Enumerative combinatorics