One of the many interpretations of the Catalan numbers
is that they count the number of noncrossing partitions of the set
A partition of
is a disjoint collection of sets whose union is
; often these subsets are called blocks. Two blocks
are crossing if they contain elements
. (These situations are not crossing:
.) A noncrossing partition is a partition with no crossing pair of blocks.
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.