V. CLUSTER NUMBER COUNTS A. Interaction proportional to the energy density of DE ( ξ$_{1}$ = 0 , ξ$_{2}$ = 0 ) In this section we present numerical results on the cluster number counts due to different interactions between DE and DM. We limit our attention to three commonly used forms of interaction between dark sectors, with the interaction proportional to the energy densities of DE, DM and total dark sectors respectively. In our numerical calculation, we consider that all models have the same σ$_{8}$ = 0 . 8 . In the end of this section we also discuss the results by normalizing the halo abundance to the present value. When the interaction is proportional to the DE density, it was found that the curvature perturbation is always stable for both quintessence and phantom DE EOS [8]. When we consider DE EoS w > − 1, we have the numerical results as shown in Fig 1. In the case when DE distribution is homogeneous, the results are shown in solid lines. With positive coupling ( ξ$_{2}$ > 0 DE decays to DM), we see in Fig 1a that the critical mass density decreases compared with the ΛCDM model. This means that with the positive coupling, cluster can be formed easier than that of ΛCDM model. However when the coupling is negative, with DM decays to DE, δ$_{c}$ is higher than that of ΛCDM, which means that it is harder for the cluster to be formed. The impact of the coupling between dark sectors is significant in causing the difference in the cluster number counts. For positive coupling ( ξ$_{2}$ > 0 DE decays to DM), the cluster number counts are bigger than that of ΛCDM model, which is clearly shown in Fig 1b-Fig 1d. However when the coupling is negative( ξ$_{2}$ < 0 with DM decays to DE), the situation is opposite and consequently cluster number counts are smaller than that of ΛCDM model. When the DE is distributed inhomogeneously, we need to consider the DE perturbation and its participation in the structure formation. Results are shown in dotted lines in Fig 1. We see that δ$_{c}$ is slightly larger than the results without DE perturbation. The difference brought by the DE inhomogeneity is really small if compared with that of the interaction. For the DE EoS w < − 1, the numerical results are shown in Fig 2. When the DE distribution is homogeneous, the influence brought by the interaction between dark sectors on the evolution of critical overdensity δ$_{c}$ and the galaxy number counts agrees well with the situation when w > − 1. However when we consider the DE is distributed inhomogeneously, the model with clustering DE with w < − 1 has a bit more number counts than the homogeneous DE distribution appeared in the large mass bin, although the difference is very small especially for the case of small 13