  An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set
such that each partition has $k$ nonempty parts and no part in any partition is
a subset of a part in a different partition. The maximum number of partitions
in an $(n,k)$-Sperner partition system is denoted $\mathrm{SP}(n,k)$. In this
paper we introduce a new construction for Sperner partition systems based on a
division of the ground set into many equal-sized parts. We use this to
asymptotically determine $\mathrm{SP}(n,k)$ in many cases where $\frac{n}{k}$
is bounded as $n$ becomes large. Further, we show that this construction
produces a Sperner partition system of maximum size for numerous small
parameter sets $(n,k)$. By extending a separate existing construction, we also
establish the asymptotics of $\mathrm{SP}(n,k)$ when $n \equiv k \pm 1
\pmod{2k}$ for almost all odd values of $k$.
