Periodicities of partition functions and Stirling numbers modulo \(p\) (Q1085193)

Let \(p(n,k)\) be the number of partitions of \(n\) into parts \(\leq k\), then the sequence \(\{p(j,k)\}^{\infty}_{j=k}\) is periodic modulo a prime \(p\). The authors obtain the minimum period \(Q(k,p)\) of this sequence and more generally the minimum period modulo \(p\) of \(\{p(n,T)\}\), the number of partitions of \(n\) whose parts lie in a fixed set \(T\) of positive integers. The minimum period modulo \(p\) of \(\{S(n,k)\}^{\infty}_{n=k}\), the sequence of Stirling numbers of the second kind, is also derived. The method uses cyclotomic polynomials over \(\mathbb Z_ p[x]\).