Editing Noncrossing partition (section)

==Definition==

A [[partition of a set]] ''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'' = { 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.

{|
|Crossing:     
|''a x b y''
|-
|Noncrossing:
|''a x y b''
|-
|Noncrossing:
|''a b x y''
|}

Equivalently, if we label the vertices of a regular ''n''-gon with the numbers 1 through ''n'', the [[convex hull]]s 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 <math>\text{NC}(S)</math>. There is an obvious order isomorphism between <math>\text{NC}(S_1)</math> and <math>\text{NC}(S_2)</math> for two finite sets <math> S_1,S_2</math> with the same size. That is, <math>\text{NC}(S)</math> depends essentially only on the size of <math> S</math> and we denote by <math>\text{NC}(n)</math> the non-crossing partitions on ''any'' set of size ''n''.