Stirling Numbers of the First Kind
The Stirling numbers of the first kind, or Stirling cycle numbers, denoted or , count the number of ways to permute a set of elements into cycles. This Demonstration illustrates the different permutations that a Stirling cycle number counts.
Snapshot 1: There is only one way to permute a list containing elements into (singleton) cycles, and therefore .
Snapshot 2: Rotating the elements in a cycle so that the last becomes the first results in the same cycle: is the same cycle as . Because of this, it is often desirable to choose a standard representation of any cycle, such as rotating it so that its greatest element is listed first. After fixing the position of the greatest element in a list of items, there are ways to permute the remaining elements to create different cycles, which means that .
Snapshot 3: The Stirling numbers of the first kind can be computed recursively; by comparing snapshot 2 and snapshot 3, it is clear that is related to .