Stirling Numbers of the First Kind

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

Contributed by: Robert Dickau (March 2011)
Open content licensed under CC BY-NC-SA



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 .

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.