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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 $n\leq 5$.