Stirling Numbers of the First Kind

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 s(n, k), 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:

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:

5 elements, one cycle

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

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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).

Copyright © 1996–2017 by Robert Dickau.

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