Periodicities of partition functions and stirling numbers modulo p

Abstract

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 p [ x].