The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating function ${\mathsf{Z}}_d(t)$ of the motives $[{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^d({\mathbb{A}}^n{)}_0]\in K_0({\mathrm{V}\mathrm{a}\mathrm{r}}_{\mathbb{C}})$ of punctual Hilbert schemes, summing over $n$, for fixed $d>0$. The result is an expression for ${\mathsf{Z}}_d(t)$ as the product of the zeta function of ${\mathbb{P}}^{d-1}$ and a polynomial ${\mathsf{P}}_d(t)$, which in particular implies that ${\mathsf{Z}}_d(t)$ is a rational function. Moreover, we reduce the complexity of ${\mathsf{P}}_d(t)$ to the computation of $d-8$ initial data, and therefore give explicit formulas for ${\mathsf{Z}}_d(t)$ in the cases $d\le 8$, which in turn yields a formula for $[{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^{\le 8}(X)]$ for any smooth variety $X$. We perform a similar analysis for the Quot scheme of points, obtaining explicit formulas for the full generating function (summing over all ranks and dimensions) for $d\le 4$. In the limit $n\to \mathrm{\infty }$, we prove that the motives $[{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^d({\mathbb{A}}^n{)}_0]$ stabilise to the class of the infinite Grassmannian $\mathrm{G}\mathrm{r}(d-1,\mathrm{\infty })$. Finally, exploiting our geometric methods, we conjecture (and partially confirm) a structural result on the ‘error’ measuring the discrepancy between the count of higher dimensional partitions and MacMahon’s famous guess.