Stirling numbers

Second kind (Stirling partition number)

The Stirling numbers of the second kind, written $S(n,k)$ or $\{\genfrac{}{}{0ex}{}{n}{k}\}$ or with other notations, count:

• The number of ways to partition a set of $n$ labelled objects into $k$ nonempty unlabelled subsets.
• The number of different equivalence relations with precisely $k$ equivalence classes that can be defined on an $n$ element set.

k n	0	1	2	3	4	5	6	7	8	9	10
0	1
1	0	1
2	0	1	1
3	0	1	3	1
4	0	1	7	6	1
5	0	1	15	25	10	1
6	0	1	31	90	65	15	1
7	0	1	63	301	350	140	21	1
8	0	1	127	966	1701	1050	266	28	1
9	0	1	255	3025	7770	6951	2646	462	36	1
10	0	1	511	9330	34105	42525	22827	5880	750	45	1

Relation to Bell numbers

Since the Stirling number $\{\genfrac{}{}{0ex}{}{n}{k}\}$ counts set partitions of an $n$-element set into $k$ parts, the sum $B_n={\sum }_{k=0}^n\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}$ over all values of $k$ is the total number of partitions of a set with $n$ members.