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Stirling Number of the First Kind

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The signed Stirling numbers of the first kind are variously denoted s(n,m) (Riordan 1980, Roman 1984), S_n^((m)) (Fort 1948, Abramowitz and Stegun 1972), S_n^m (Jordan 1950). Abramowitz and Stegun (1972, p. 822) summarize the various notational conventions, which can be a bit confusing (especially since an unsigned version S_1(n,m)=|s(n,m)| is also in common use). The signed Stirling number of the first kind s(n,m) is are returned by StirlingS1[n, m] in the Wolfram Language, where they are denoted S_n^((m)).

The signed Stirling numbers of the first kind s(n,m) are defined such that the number of permutations of n elements which contain exactly m permutation cycles is the nonnegative number

|s(n,m)|=(-1)^(n-m)s(n,m).
(1)

This means that s(n,m)=0 for m>n and s(n,n)=1. A related set of numbers is known as the associated Stirling numbers of the first kind. Both these and the usual Stirling numbers of the first kind are special cases of a general function d_r(n,k) which is related to the number of cycles in a permutation.

The triangle of signed Stirling numbers of the first kind is

1 -1 1 2 -3 1 -6 11 -6 1 24 -50 35 -10 1
(2)

(OEIS A008275). Special values include

s(n,0)=delta_(n0)
(3)
s(n,1)=(-1)^(n-1)(n-1)!
(4)
s(n,2)=(-1)^n(n-1)!H_(n-1)
(5)
s(n,3)=1/2(-1)^(n-1)(n-1)![H_(n-1)^2-H_(n-1)^((2))]
(6)
s(n,n-1)