A combinatorial proof of non-speciality of systems with at most 9 imposed base points

It is known that the Segre-Gimigliano-Harbourne-Hirschowitz Conjecture holds for linear systems of curves with at most 9 imposed base fat points. We give a nice proof based on a combinatorial method of showing non-speciality of such systems. We will also prove, by the same method, that systems $mathcal L(km;m^{ imes k^2})$ and $mathcal L(km+1;m^{ imes k^2})$ are non-special.