Stirling Numbers of the Second Kind

number of points

4

number of partitions

2

diagram

1

The Stirling numbers of the second kind, or Stirling partition numbers, sometimes denoted



n

k



, count the number of ways to partition a set of

n

elements into

k

discrete, nonempty subsets. This Demonstration illustrates the different partitions that a Stirling partition number counts. The sums of the Stirling partition numbers are the Bell numbers.

Details

Snapshot 1: there is only one way to partition

n

elements into

n

nonempty subsets, and therefore



n

=1

Snapshot 2: similarly, there is only one way to partition

n

elements into 1 nonempty subset, which means that



n

1

=1

Snapshot 3: the Stirling numbers of the second kind can be computed recursively; by comparing Snapshot 2 and Snapshot 3, it is apparent that



5

1



and



6

2



are related

External Links

Stirling Number of the Second Kind (Wolfram MathWorld)

Small Set Partitions

Stirling's Triangles

Permanent Citation

Robert Dickau

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