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Definition & Example
 A set partition of size $n$ is a partition of the set $\mathcal{S} = \{1,\ldots,n\}$. This is a collection of nonempty pairwise disjoint subsets (parts) of $\mathcal{S}$ whose union is $\mathcal{S}$. In symbols, $\mathcal{P} = \{P_1,\ldots,P_k \}$ such that
the 5 Set partitions of size 3  
{{1,2,3}}  {{1,2},{3}}  {{1,3},{2}}  {{1},{2,3}}  {{1},{2},{3}} 

Set partitions of size $n$ are graphically represented by drawing the numbers $1$ through $n$ around a circle and then drawing the convex hulls of the blocks.

The number of set partitions of size $n$ is $n$th Bell number $B_n$ (A000110).

The number of set partitions of size $n$ into $k$ blocks is the Stirling number of the second kind (A008277).
Additional information

The set partitions of size $n$ form a poset by containment order. This poset is indeed a lattice which is the intersection lattice of the braid arrangement.

A set partition is said to be noncrossing if the graphical representation does not have any crossing blocks. In symbols, this is to say that there does not exist $P_i, P_j \in \mathcal{P}$ which contains elements $a, b \in P_i$ and $x, y \in P_j$ such that $a < x < b < y$. The number of noncrossing set partitions of size $n$ is the nth Catalan number.
References
Sage examples
Technical information for database usage
 A set partition is represented as a set of disjoint blocks, which are themselves sets.
 Set partitions are graded by the size of the ground set.
 The database contains all set partitions of size at most 7.