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 (Var ₂) 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 P^d-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.