On a partition problem of Canfield and Wilf

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a is in M for only finitely many a. Denote by p_{A,M}(n) the number of partitions of n with parts in A and multiplicities in M. It is proved that there exist infinite sets A and M of positive integers whose partition function p_{A,M} has weakly superpolynomial but not superpolynomial growth. The counting function of the set A is A(x) = \sum_{a \in A, a\leq x} 1. It is also proved that p_{A,M} must have at least weakly superpolynomial growth if M is infinite and A(x) >> log x.

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