Partitions with k sizes from a set

Abstract

Let n and k be two positive integers and let A be a set of positive integers. We define t(A)(n, k) to be the number of partitions of n with exactly k sizes and parts in A. As an implication of a variant of Newton's product-sum identities we present a generating function for t(A)(n, k). Subsequently, we obtain a recurrence relation for t(A)(n, k) and a divisor-sum expression for t(A)(n, 2). Also, we present a bijective proof for the latter expression.