We have

$$\log f_{\mathfrak{d}}=\sum_{\beta}\sum_{\bf P}k(\beta)N_{\bf P}t^{\sum_{i}|{\bf P}_{i}|}z^{-k(\beta)m_{\mathrm{out}}},$$

where the sum is over all $\beta\in H^{2}(X,\mathbb{Z})$ with $\beta\cdot D_{i}\geq 0$, $k(\beta):=\beta\cdot D_{\mathrm{out}}>0$, and partitions ${\bf P}$ with $|{\bf P}_{i}|=\beta\cdot D_{i}$.