ON A PARTITION PROBLEM OF CANFIELD AND WILF

摘要

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 = ! a∈A maa where ma ∈ M ∪ {0} for all a ∈ A, and ma ∈ M for only finitely many a. Denote by pA,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 pA,M has weakly superpolynomial but not superpolynomial growth. The counting function of the set A is A(x) = ! a∈A,a≤x 1. It is also proved that pA,M must have at least weakly superpolynomial growth if M is infinite and A(x) # log x.