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arXiv · 1207.3317

Restricted integer partition functions

Noga Alon
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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 $A=\{k^k\}_{k \geq 1}$, $0 n_0$. This answers a question of Ljujić and Nathanson.

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Noga Alon. 2012-07-13. Restricted integer partition functions. https://arxiv.org/abs/1207.3317

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