 $r_i$ is either $0$ or $0$ for all $i$, and any $\alpha=(\alpha_0,\ldots,\alpha_m)$ in $A(n_0, \ldots, n_m)$ put $$u_{g,\alpha}^{(n_0^{r_0},\ldots,n_m^{r_m})} :=  I_g \cdot \sum_{(h,f) \in P_g} \prod_{i=0}^m \tau_{n_i} \bigl(h(\alpha_i),f(\alpha_{i}) \bigr)^{r_{i}}$$ and define $$u_{g}^{(n_0^{r_0},\ldots,n_m^{r_m})}