  We give an explicit stratification of the punctual Hilbert schemes of $n$
points of $\mathbb{A}^{m+1}$ with respect to $m$-dimensional partitions in the
Grothendieck group of varieties. As an application, we calculate the classes of
the punctual Hilbert schemes of $n$ points of $\mathbb{A}^3$ and the classes of
the Hilbert schemes of $n$ points of $\mathbb{A}^3$ in the Grothendieck of
varieties for $n\leq 5$.
