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Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the latticePart$(n)$ of all partitions of an $n$-element set has a four-element generatingset. In 1983, L. Z\'adori gave a new proof of this fact with a particularlyelegant construction. Based on his construction from 1983, the present papergives a lower bound on the number $\nu(n)$ of four-element generating sets ofPart$(n)$. We also present a computer assisted statistical approach to $\nu(n)$for small values of $n$.  In his 1983 paper, L. Z\'adori also proved that for $n\geq 7$, the latticePart$(n)$ has a four element generating set that is not an antichain. He leftthe problem whether such a generating set for $n\in\{5,6\}$ exists open. Herewe solve this problem in negative for $n=5$ and in affirmative for $n=6$.  Finally, the main theorem asserts that the direct product of some powers ofpartition lattices is four-generated. In particular, by the first part of thistheorem, Part$(n_1)\times$ Part$(n_2)$ is four-generated for any two distinctintegers $n_1$ and $n_2$ that are at least 5. The second part of the theorem istechnical but it has two corollaries that are easy to understand. Namely, thedirect product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(3n-14)$is four-generated for each integer $n\geq 9$. Also, for every positive integer$u$, the $u$-th the direct power of the direct product Part$(n)$ $\times$Part$(n+1)$ $\times\dots\times$ Part$(n+u-1)$ is four-generated for all butfinitely many $n$. If we do not insist on too many direct factors, then theexponent can be quite large. For example, our theorem implies that the$10^{127}$-th direct power of Part$(1011)$ $\times$ Part$(1012)$ $\times \dots\times$ Part$(2020)$ is four-generated.

Four-element generating sets of partition lattices and their direct products