Multifactorisations and Divisor Functions

Abstract

We consider a joint ordered multifactorisation for a given positive integer $n\geq 2$ into $m$ parts, where $n=n_1~\times~\ldots~\times~n_m$, and each part $n_j$ is split into one or more component factors. Our central result gives an enumeration formula for all such joint ordered multifactorisations $\mathcal{N}_m(n)$. As an illustrative application, we show how each such factorisation can be used to uniquely construct and so count the number of distinct additive set systems (historically referred to as complementing set systems). These set systems under set addition generate the first $n$ non-negative consecutive integers uniquely and, when each component set is centred about 0, exhibit algebraic invariances. For fixed integers $n$ and $m$, invariance properties for $\mathcal{N}_m(n)$ are established. The formula for $\mathcal{N}_m(n)$ is comprised of sums over associated divisor functions and the Stirling numbers of the second kind, and we conclude by deducing sum over divisor relations for our counting function $\mathcal{N}_m(n)$.