> [!NOTE] Definition > For natural numbers $n,k$, the number of partitions of $[k]=\{ 1,2,\dots,k \}$, denoted $B_{k}$, is known as a Bell number. # Properties **Relation to Stirling numbers of the second kind**: By sorting the partitions of $[k]$ by the number of parts, say $n$, we get $B_{k}=\sum_{n=0}^{k} S(k,n)$where $S(n,k)$ denotes a [[Stirling numbers of the second kind|Stirling number of the second kind]].