Associated Stirling Number of the First Kind

The associated Stirling numbers of the first kind are defined as the number of permutations of a given number having exactly permutation cycles, all of which are of length or greater (Comtet 1974, p. 256; Riordan 1980, p. 75). They are a special case of the more general numbers , and have the recurrence relation

(1)

with initial conditions for , and (Appell 1880; Tricomi 1951; Carlitz 1958; Comtet 1974, pp. 256, 293, and 295). The generating function for is given by

(2)

(Comtet 1974, p. 256). The associated Stirling numbers of the first kind satisfy the sum identity

(3)

For and a prime,

(4)

For all integers ,

(5)

and similarly,

(6)

(Comtet 1974, p. 256).

Special cases of the associated Stirling numbers of the first kind are given by

			(7)
			(8)
			(9)
			(10)

(Comtet 1974, p. 256). The triangle of these numbers is given by

(11)

(OEIS A008306).