The motive of the Hilbert scheme of points in all dimensions

We prove a closed formula for the generating series of the motives in of punctual Hilbert schemes, summing over , for fixed . The result is an expression for as the product of the zeta function of and a polynomial , which in particular implies that is a rational function. Moreover, we reduce the complexity of to the computation of initial data, and therefore give explicit formulas for in the cases , which in turn yields a formula for for any smooth variety , 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 . In the limit , we prove that the motives stabilise to the class of the infinite Grassmannian . 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.