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The lattice of noncrossing partitions of a finite set forms a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.

The noncrossing partition lattice has taken on importance because of its role in free probability theory.

Counting partitions

The total number of partitions of an n-element set is the Bell number B_n. The first several Bell numbers are B₀ =

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