There are two kinds of Stirling numbers: Stirling numbers of the first kind and Stirling numbers of the second kind. They appear in many situations in combinatorics.
Stirling Numbers of the First Kind
For example, because (writing all our permutations in cycle notation) we have the permutations .
Stirling Numbers of the Second Kind
For example, because we have the partitions .
Stirling numbers of the second kind satisfy the recursion
(This can easily be shown by considering the cases that is in a part of size or size larger than .) They can also be calculated using the formula
The total number of partitions of an -element set is and is called the th Bell number.