  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 $n=\sum_{a \in A} m_a a$ where $m_a \in M \cup {0}$ for all $a$, and all
numbers $m_a$ 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 $n_0$ so that for $A={k!}_{k \geq
1}$ or for $A=\{k^k\}_{k \geq 1}$, $0<p(n,A,M) \leq n^c$ for all $n>n_0$. This
answers a question of Ljuji\'c and Nathanson.

Combinatorics