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## Tuesday, April 20, 2010

### Stirling numbers of the second kind

( Repeat / memorized ) from Aigner, A course in enumeration.

Stirling numbers of the second kind: S(n,k).
Definition. S(n,k) is the number of k-partitions of an n-set.

Example:
{A}
S(1,1)=1
A

{A,B}
S(2,1)=1
AB
S(2,2)=1
A-B

{A,B,C}
S(3,1)=1
ABC
S(3,2)=3
A-BC
B-AC
C-AB
S(3,3)=1
A-B-C

Almost similar as for the number of combinations ( C(n,k)= C(n-1,k-1) + C(n-1,k) ) we can define the Stirling numbers of the second kind recursively:
S(n,k) = S(n-1,k-1) + k * S(n-1,k).
This recursion is easy to understand as we will see when we continue with the examples for {A,B,C,D}.

/* To be continued */

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)