Stirling Numbers of the First Kind

number of points

3

number of cycles

1

diagram

1

The Stirling numbers of the first kind, or Stirling cycle numbers, denoted

s(n,k)

or

(k)

S

n

, count the number of ways to permute a set of

n

elements into

k

cycles. This Demonstration illustrates the different permutations that a Stirling cycle number counts.

Details

Snapshot 1: There is only one way to permute a list containing

n

elements into

n

(singleton) cycles, and therefore

s(n,n)=1

.

Snapshot 2: Rotating the elements in a cycle so that the last becomes the first results in the same cycle:

{1,2,3}

is the same cycle as

{3,1,2}

. 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

n

items, there are

(n-1)!

ways to permute the remaining

n-1

elements to create different cycles, which means that

s(n,1)=(n-1)!

.

Snapshot 3: The Stirling numbers of the first kind can be computed recursively; by comparing snapshot 2 and snapshot 3, it is clear that

s(5,1)

is related to

s(6,2)

.

External Links

Stirling Number of the First Kind (Wolfram MathWorld)

Stirling Numbers of the Second Kind

Small Set Partitions

Stirling's Triangles

Permanent Citation

Robert Dickau

"Stirling Numbers of the First Kind"
http://demonstrations.wolfram.com/StirlingNumbersOfTheFirstKind/
Wolfram Demonstrations Project
Published: March 7, 2011