The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating series (Formula presented.) of the motives (Formula presented.) in (Formula presented.) of punctual Hilbert schemes, summing over (Formula presented.), for fixed (Formula presented.). The result is an expression for (Formula presented.) as the product of the zeta function of (Formula presented.) and a polynomial (Formula presented.), which in particular implies that (Formula presented.) is a rational function. Moreover, we reduce the complexity of (Formula presented.) to the computation of (Formula presented.) initial data, and therefore give explicit formulas for (Formula presented.) in the cases (Formula presented.), which in turn yields a formula for (Formula presented.) for any smooth variety (Formula presented.), providing infinite families of new examples of motives of singular Hilbert schemes. 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 (Formula presented.). In the limit (Formula presented.), we prove that the motives (Formula presented.) stabilise to the class of the infinite Grassmannian (Formula presented.). Finally, exploiting our geometric methods, we propose a structural formula on the ‘error’ measuring the discrepancy between the count of higher dimensional partitions and MacMahon's famous guess.