This web form gives a solution to the problem for any r and m (for the safety of the departmental server, m and r are allowed only to be only 20 or less, but the programs can be downloaded from http://www.math.unl.edu/~bharbourne1/GMS/ and run for any r and m within the defined range for integers for your version of C).

The script does not do a calculation involving an ideal for some actual set of points; it applies a theorem. Thus the result does not give experimental data; it gives the actual hilbert functions.

Note that not every configuration type can occur over every algebraically closed field. Sometimes it depends on the characteristic. This web form ignores that issue, so if you have a particular characteristic in mind you will have to ignore hilbert functions for types that do not occur in the characteristic you're interested in. (An ArXiv posting will be made soon by Geramita/Harbourne/Migliore that indicates which types occur in characteristics.) To learn more abut configuration types, here is a pdf file of a talk I gave recently.

The easiest way to specify the hilbert function of I(mZ) is to specify m, n and the type of the points Z.