Restricted integer partition functions

Abstract

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 $A={k!}_{k\ge 1}$ or for $A=\{k^k{\}}_{k\ge 1}$, $0<p(n,A,M)\le n^c$ for all $n>n_0$. This answers a question of Ljujić and Nathanson.