The numerical results by choosing this coupling are shown in Fig 3. The solid lines are for the homogeneous distribution of DE. For small coupling, the lines are close to that of ΛCDM result. We observe that more positive coupling leads to more number counts of clusters Fig 3c. This can be understood from δ$_{c}$ and also the ratio δ$_{c}$ ( z ) /σ$_{8}$D ( z ), which are smaller at high redshift for more positive ξ$_{1}$ . This result is also consistent with the case when the interaction is proportional to the DE energy density. The dotted lines are for the situations when DE is distributed inhomogeneously and we need to consider the DE perturbation and its participation in the structure formation. From the behavior of δ$_{c}$ we see that δ$_{c}$ is suppressed at low redshift. This suppression is more obvious than the case of homogeneous DE. In the ratio δ$_{c}$ ( z ) /σ$_{8}$D ( z ) shown in Fig 3b, the ratio due to the inhomogeneous DE is smaller than that of the homogeneous DE. This leads to a larger cluster abundance due to the inhomogeneous DE distribution than that of homogeneous DE distribution as shown in Fig 3c. Here, again, the curvature perturbation can only be stable when DE EOS is smaller than -1 [8] and the coupling has to be positive [18]. The numerical results are shown in Fig 4. The results are similar to the case when the coupling is proportional to the energy density of DM. With more positive coupling, the critical overdensity is smaller which leads to the bigger galaxy number counts than that of the ΛCDM model. For the same coupling, if the DE is clustering, the critical overdensity δ$_{c}$ and δ$_{c}$ ( z ) /σ$_{8}$D ( z ) will be suppressed at small redshift. Therefore it is observed in Fig 4c,Fig 4d that at small the inhomogeneous DE distribution leads to a larger cluster number abundance as compared with the homogeneous DE distribution. So far in the above discussions we have fixed σ$_{8}$ = 0 . 8 as given by present observations [42]. Considering that at redshift zero all models have the same number density of halos, the normalization is required to be done by adjusting σ$_{8}$ in each model such that δ$_{c}$/σ$_{8}$ is equal to the fiducial σ$_{8}$ = 0 . 8 ΛCDM model We dedicate the rest of this section to discuss the implications on our results by normalizing models to the same halo abundance at z = 0. It is interestingly observed that the qualitative structure of curves changes dramatically depending on the choices of the normalizations, which was also found in studying the number counts in homogeneous 16 C. Interaction proportional to the total energy density of dark sectors ( ξ = ξ$_{1}$ = ξ$_{2}$ > 0 ) D. Normalization σ 0 8 ,model = δ$_{c,model}$ ( z = 0) δ$_{c,}$$_{Λ}$ ( z = 0) σ 0 8 , Λ . (46)