Restricted integer partition functions

TL;DR

This paper investigates restricted integer partition functions, demonstrating the existence of infinite sets with unique partitions and establishing bounds for specific sets of parts, thereby resolving several open problems in the field.

Contribution

It constructs infinite sets of parts and multiplicities that produce unique partitions for all positive integers and provides bounds for partitions with specific infinite sets of parts.

Findings

Existence of infinite sets A and M with p(n,A,M)=1 for all n

Bounded partition counts for A={k!} and A={k^k} with an infinite M

Resolution of open problems by Canfield, Wilf, Ljujić, and Nathanson

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 $A={k!}_{k \geq 1}$ 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.