On a partition problem of Canfield and Wilf

TL;DR

This paper investigates a specific partition problem involving sets of positive integers, demonstrating the existence of infinite sets with partition functions exhibiting weakly superpolynomial growth, and establishing growth conditions based on set properties.

Contribution

It proves the existence of infinite sets A and M with partition functions that grow weakly superpolynomially but not superpolynomially, and establishes growth conditions related to the set A.

Findings

Existence of infinite sets A and M with weakly superpolynomial partition function growth.

Partition function p_{A,M} can grow weakly superpolynomially without being superpolynomial.

Growth of p_{A,M} is at least weakly superpolynomial if M is infinite and A(x) >> log x.

Abstract

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.