The diffusion and coalescence of individual atoms on a nanostructured surface are treated in a purely statistical way. From this, analytical formulas are derived which, from a known initial state, give the final cluster size distribution on a surface after the diffusion of all the individual atoms. Unexpectedly, it turns out that these formulas allow obtaining a statistical law giving the size histogram of the clusters when only homogeneous nucleation occurs with a critical germ equal to 2, and in the situation where nucleation starts once the deposition of the atoms is completed.