**Table 1.** The range of μ in terms of on-site interaction, *U* and nearest-neighbour exchange *V* for different incompressible phases. The discussion is restricted until the MI (2) lobe.

**Abstract**

We investigate the phase diagram of interacting bosons in a tripartite lattice in two dimensions. An analytic computation of the phase diagram in the parameter space defined by the on-site boson–boson repulsion parameter and the chemical potential is done via a second-order strong coupling perturbation theory on the extended Bose–Hubbard model, which is hence supplemented by numerical mean-field calculations. Interesting results emerge in the form of exotic density ordered phases with one-third, two-third and unity filling of bosons per trimer. The pattern repeats as the chemical potential is increased. Further regions of supersolid and superfluid phases are noted in the numerical study of the mean-field model where the phase diagram shows a transition from compressible to insulating phases.