Noncrossing partition

There are 42 noncrossing and 10 crossing partitions of a 5-element set
The 14 noncrossing partitions of a 4-element set ordered by refinement in a Hasse diagram

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 number of noncrossing partitions of a set of n elements is the nth Catalan number. The number of noncrossing partitions of an n-element set with k blocks is found in the Narayana number triangle.