Noncrossing Partitions
Noncrossing Partitions
One of the many interpretations of the Catalan numbers is that they count the number of noncrossing partitions of the set .
C
n
[n]={1,2,…,n}
A partition of is a disjoint collection of sets whose union is ; often these subsets are called blocks. Two blocks and are crossing if they contain elements and such that . (These situations are not crossing: , .) A noncrossing partition is a partition with no crossing pair of blocks.
[n]
[n]
A
W
a,b∈A
x,y∈W
a<x<b<y
a<b<x<y
a<x<y<b
This Demonstration shows two figures related to noncrossing partitions: one in which points are arranged at the corners of a regular polygon, with line segments connecting members of the same block; and one in which points are arranged in a line, with arcs joining members of the same block.