On the Number of Multiplicative Partitions of a Multi-partite Number

A \(t\)-partite number is a \(t\)-tuple \(\vec{n} = (n_1, \ldots, n_t)\), where \(n_1, \ldots, n_t\) are positive integers. For a \(t\)-partite number \(\vec{n}\), let \(f_t(\vec{n})\) be the number of different ways to write \(\vec{n}\) as a product of \(t\)-partite numbers, where the multiplication is performed coordinate-wise, \((1, 1, \ldots, 1)\) is not used as a factor of \(\vec{n}\), and two factorizations are considered the same if they differ only in the order of the factors. This paper gives the following explicit upper bound for the multiplicative partition function \(f_t(\vec{n})\):
\[f_t(n_1, \ldots, n_t) \leq M^{w(t)},\, \text{where}\,\, M = \Pi_{i=1}^t n_i \,\,\text{and}\,\, w(t) = \frac{\log((t+1)1)}{t\log2}\].