  A vector space partition of $\mathbb{F}_q^v$ is a collection of subspaces
such that every non-zero vector is contained in a unique element. We improve a
lower bound of Heden, in a subcase, on the number of elements of the smallest
occurring dimension in a vector space partition. To this end, we introduce the
notion of $q^r$-divisible sets of $k$-subspaces in $\mathbb{F}_q^v$. By
geometric arguments we obtain non-existence results for these objects, which
then imply the improved result of Heden.
