One of the many interpretations of the Catalan numbers *C _{n}* is that they count the number of noncrossing partitions of the set {1, 2, ...,

One representation of noncrossing partitions is one in which points are arranged at the corners of a regular polygon, with line segments connecting members of the same block.

Two points, two non-crossing partitions {{1, 2}}—joined by a line since they’re in the same subset—and {{1}, {2}}—not joined since they’re in different subsets:

Three points, five non-crossing partitions ({{1, 2, 3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, and {{1},{2},{3}}):

Four points, 14 non-crossing partitions:

Five points, 42 non-crossing partitions:

An equivalent representation is one in which points are arranged in a line, with arcs joining members of the same block.

Two points, two non-crossing partitions ({{1, 2}} and {{1}, {2}}, again):

Three points, five non-crossing partitions ({{1, 2, 3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, and {{1},{2},{3}}):

Four points, 14 non-crossing partitions:

Five points, 42 non-crossing partitions:

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