Periodicities of partition functions and stirling numbers modulo p

If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}_{n ≥ 0}, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Z_px].