Given an integer n, the three flag Hilbert scheme is the variety that parametrizes flags of nested subschemes of the affine plane that are supported on only one point and that have length, respectively, n, n+1 and n+2. We will show that these varieties admit an affine paving given by attracting cells for a natural torus action. This, in turns, allows the computation of the homology groups and the description of their Poincaré polynomials in a combinatorial way. Moreover it possible to find a generating function for all such Poincaré polynomials as we consider all non negative integers. Similar results for the standard Hilbert scheme (only one subscheme at the time) and for the two flag Hilbert scheme, are classical results, of, respectively, Goettsche (for the generating function) + Ellingsrud-Stromme (for the paving and the homology) and Cheah.
