We investigate the interactions between substitutional carbon atoms on the defect free, $(2\xd71)$ reconstructed Si(001) surface, and bring evidence that the interaction energy differs significantly from the inverse-cube distance dependence that is predicted by the theory of force dipoles on an elastic half-space. Based on Tersoff potentials, we also calculate the interactions between carbon atoms and dimer vacancies. The calculations indicate that dimer vacancies (DVs) are strongly stabilized by fourth-layer C atoms placed directly underneath them. By use of simple model Monte Carlo simulations, we show that the computed interactions between carbon atoms and DVs lead to self-assembled vacancy lines, in qualitative agreement with recent experimental results.