Congruence Problems Involving Stirling Numbers of the First Kind

It is by now well known that the parities of the binomial coefficients show a fractal-like appearance when plotted in the x-y plane. Similarly, if f(n,k) is some counting sequence and/? is a prime, we can plot an asterisk at (//,&) iff(n,k) # 0 (mod/?), and a blank otherwise, to get other complex, and often interesting, patterns. For the ordinary and Gaussian binomial coefficients and for the Stirling numbers of the second kind, formulas for the number of asterisks in each column are known ([12], [2], [4], [1]). Moreover, in each row the pattern is periodic, and formulas for the minimum period have been bound ([2], [3], [6], [7], [12], [13], [9], [15]) in all three cases. If KM, the signless Stirling number of the first kind, denotes, as usual, the number of permutations of/? letters that have k cycles, then for fixed k and/? we will show that there are only finitely many n for which m =£ 0 (mod/?), i.e., there are only finitely many asterisks in each row of the pattern. Let v(n,k) be the number of these. To describe the generating function of the v(n,k) we first need to define a special integer modulo p. We say that a nonnegative integer n is special modulo/? if