Stirling number of the first kind

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The number s(n, r) of ways of partitioning a set of n elements into r cycles. For example, the set {1, 2, 3, 4} can be partitioned into two cycles in the following ways:

So s(4, 2)=11. Clearly s(n, 1)=(n−1)! and s(n, n)=1. It can be shown that

s(n+1, r)=s (n, r−1)+ns(n, r)

Some authors define these numbers differently so that they satisfy s(n+1, r)=s(n, r−1)−ns(n, r). The result is that the values are the same except that some of them occur with a negative sign.

Rather like the binomial coefficients, the Stirling numbers occur as coefficients in certain identities. They are named after the Scottish mathematician James Stirling (1692–1770).

Subjects: Mathematics.

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