Noncrossing partition

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 set of all noncrossing partitions is one of many sets enumerated by the Catalan numbers. The number of noncrossing partitions of an n-element set with k blocks is found in the Narayana number triangle.

A partition of a set S is a pairwise disjoint set of non-empty subsets, 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 = { 1, ..., n }. 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 { { a, b }, { x, y } } 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 are denoted ${\displaystyle {\text{NC}}(S)}$. There is an obvious order isomorphism between ${\displaystyle {\text{NC}}(S_{1})}$ and ${\displaystyle {\text{NC}}(S_{2})}$for two finite sets ${\displaystyle S_{1},S_{2}}$ with the same size. That is, ${\displaystyle {\text{NC}}(S)}$ depends essentially only on the size of ${\displaystyle S}$ and we denote by ${\displaystyle {\text{NC}}(n)}$ the non-crossing partitions on any set of size n.

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