Associated Stirling Number of the First Kind

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

(1)

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

e^(-tu)(1-t)^(-u)=1+sum_(n=2)^inftysum_(k=1)^(|_n/2_|)(d_2(n,k))/(n!)t^nu^k
=1+((t^2)/2+(t^3)/3+(t^4)/4+(t^5)/5+(t^6)/6+...)u
+((t^4)/8+(t^5)/6+(13t^6)/(72)+...)u^2+((t^6)/(48)+...)u^3+...

(2)

as a formal power series, or for with fixed complex number and the principal branch of (Comtet 1974, p. 256). Here denotes the floor function. For , the associated Stirling numbers of the first kind satisfy the sum identity

(3)

For an integer and a prime,

(4)

For positive integers ,

(5)

and, for integers ,

(6)

(Comtet 1974, p. 256).

Special cases of the associated Stirling numbers of the first kind are given by (with in the first formula and positive integers in the remaining formulas)