Stirling numbers of the first kind

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The Stirling numbers of the first kind s(n, k) count the number of ways to permute a list of n items into k cycles.

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 illegible diagrams showing the cycles for permutations of a list with five elements.

s(5, 1) = 24:

s(5, 2) = 50:

s(5, 3) = 35:

s(5, 4) = 10:

s(5, 5) = 1:

Designed and rendered using Mathematica 3.0 for NeXT.