Restricted integer partition functions

For two sets A and M of positive integers and for a positive integer n, let p(n, A, M) denote the number of partitions of n with parts in A and multiplicities in M, that is, the number of representations of n in the form n = ∑a∈Am_aa where m_a ∈ M∪{0} for all a, and all numbers ma but finitely many are 0. It is shown that there are infinite sets A and M so that p(n, A, M) = 1 for every positive integer n. This settles (in a strong form) a problem of Canfield and Wilf. It is also shown that there is an infinite set M and constants c and n0 so that for A = {k!}_k≥1 or for A = {k^k}k≥1, 0 < p(n, A, M) ≤ nc for all n > n₀. This answers a question of Ljujić and Nathanson.