\sum_{ \sigma : \text{Ext}(\sigma) = \Gamma } e^{-\beta \sum_{\gamma \in \Gamma_\sigma} \Phi(\gamma)} = \prod_{\gamma^i \in \Gamma}\left[e^{-\beta \Phi(\gamma^i) } \prod_{j=0}^q \left( \sum_{\Gamma^j \in D^j(\text{Int}_j(\gamma^i))} e^{-\beta \sum_{\gamma^j \in \Gamma^j} \Phi(\gamma^j)}