Partitions with k sizes from a set

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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.