Essays/Stirling Numbers

Stirling numbers of the first kind and of the second kind can be computed as follows:

   s1=: 1: ` (<:    ((0,]) + [ * ],0:) $:@<:) @. *
   s2=: 1: ` (i.@>: ((0,]) + [ * ],0:) $:@<:) @. *

   s1"0 i.7
1   0   0   0  0  0 0
0   1   0   0  0  0 0
0   1   1   0  0  0 0
0   2   3   1  0  0 0
0   6  11   6  1  0 0
0  24  50  35 10  1 0
0 120 274 225 85 15 1
   s2"0 i.7
1 0  0  0  0  0 0
0 1  0  0  0  0 0
0 1  1  0  0  0 0
0 1  3  1  0  0 0
0 1  7  6  1  0 0
0 1 15 25 10  1 0
0 1 31 90 65 15 1

The Stirling numbers of the first and second kinds are related in a simple way:

   %. s2"0 i.7
1    0   0    0   0   0 0
0    1   0    0   0   0 0
0   _1   1    0   0   0 0
0    2  _3    1   0   0 0
0   _6  11   _6   1   0 0
0   24 _50   35 _10   1 0
0 _120 274 _225  85 _15 1

   (s1"0 i.7) -: | %. s2"0 i.7
1
   (s2"0 i.7) -: | %. s1"0 i.7
1

That is, the table of Stirling numbers of one kind are the absolute values of the matrix inverse of the table of Stirling numbers of the other kind.

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See also

• MathWorld: Stirling numbers

of the first kind and of the second kind

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Contributed by Roger Hui. s1 and s2 are slightly modified from Roger Hui and Kenneth E. Iverson, Representations of Recursion, APL95 Conference Proceedings, June 1995.