The Stirling numbers of the first kind, or Stirling cycle numbers,
count the number of ways to permute a list of *n* items into
*k* cycles. Common notations are *s*(*n*, *k*) and
,
where the first is slightly easier to type.

For example, the list {1, 2, 3, 4} can be permuted into two cycles in the following ways:

- {{1,3,2}, {4}}
- {{1,2,3}, {4}}
- {{1,4,2}, {3}}
- {{1,2,4}, {3}}
- {{1,2}, {3,4}}
- {{1,4,3}, {2}}
- {{1,3,4}, {2}}
- {{1,3}, {2,4}}
- {{1,4}, {2,3}}
- {{1}, {2,4,3}}
- {{1}, {2,3,4}}

There are 11 such permutations, thus *s*(4, 2) = 11.

Here are some diagrams showing the cycles for permutations of a list with five elements.

*s*(5, 1) = 24, that is, all 5 elements form 1 cycle:

*s*(5, 2) = 50, meaning 5 elements form 2 cycles, where a dot is a cycle of length 1:

*s*(5, 3) = 35:

*s*(5, 4) = 10:

*s*(5, 5) = 1:

Designed and rendered using *Mathematica* 3.0 (for NeXT) and—much later—7.0 (for Windows).

© 1996–2023 by Robert Dickau.

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