Stirling Numbers of the First Kind

Requires a Wolfram Notebook System

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

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.