Stirling Sequences

How many ways can you break up the numbers 1,2,3,4,5 into 3 non-empty sets?

1 2 345
1 23 45
1 24 35
1 25 34
1 234 5
1 235 4
1 245 3

12 3 45
12 34 5
12 35 4
13 2 45
13 24 5
13 25 4
14 2 35
14 23 5
14 25 3
15 2 34
15 23 4
15 24 3

123 4 5
124 3 5
125 3 4
134 2 5
135 2 4
145 2 3

So S(5,3)=25
The numbers S(n,k) are called Stirling numbers of the second kind. If we fix k and let n run through the integers, we get a stirling sequence.