Punctual Hilbert schemes of points of ${\mathbb{A}}^3$ in the Grothendieck group of varieties

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 $n\le 5$.