The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating function Zd(t) of the motives [Hilbᵈ( Aⁿ)₀] ∈ K₀(VarC) of punctual Hilbert schemes, summing over n, for fixed $d>0$. The result is an expression for Zd(t) as the product of the zeta function of Pᵈ⁻¹ and a polynomial Pd(t), which in particular implies that Zd(t) is a rational function. Moreover, we reduce the complexity of Pd(t) to the computation of $d-8$ initial data, and therefore give explicit formulas for Zd(t) in the cases d ≤ 8, which in turn yields a formula for [Hilb≤ 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 ≤ 4. In the limit n → ∞, we prove that the motives [Hilbᵈ( Aⁿ)₀] stabilise to the class of the infinite Grassmannian Gr(d-1,∞). 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.