give an intuition now. Given an injection of sets $f:[m] \hookrightarrow [n]$ we would like to obtain a map of spaces $f_{\text{*}}:\operatorname{Conf}_n^{\operatorname{sink}}(G,V) \rightarrow \operatorname{Conf}_n^{\operatorname{sink}}(G,V)$. Given no extra data, there doesn’t seem to be any obvious way to do this. However, if we are also given a map $g$ from the compliment of the image of $f$ in $[n]$ to the set of vertices $V$, then we can define such a map. Namely, the pair $(f,g)$ induces a map $(f,g)_{\text{*}}:\operatorname{Conf}_m^{\operatorname{sink}}(G,V) \rightarrow \operatorname{Conf}_n^{\operatorname{sink}}(G,V)$, where one uses $f$ to push forward the coordinates that are given, while using $g$ to determine to which vertices the newly introduced points should be sent. That is, $$(f,g)_{\text{*}}(x_1,\ldots,x_m)_j = \begin{cases} x_{f^{-1}(j)} &\text{ if $j$ is in the image of $f$}\\ g(j) &\text{ otherwise.}\end{cases}$$ This gives the homology groups $H_i(\operatorname{Conf}_{\bullet}^{\operatorname{sink}}(G,V))$ what [@R2] calls generalized representation stability. One major consequence of this stability is the following.\

\[sinkexpstab\] The modules $H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$ enjoy the following properties:

1.  there exist polynomials $p_1^{(i)},\ldots,p_{\delta_i}^{(i)} \in {\mathbb{Q}}[x]$ such that for all $n \gg 0$ $$\operatorname{rank}_{{\mathbb{Z}}}(H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))) = p_1^{(i)}(n) + p_2^{(i)}(n)2^n + \ldots + p_{\delta_i}^{(i)}(n)\delta_i^n$$ where $$\delta_i = \min\{i+1, |V|\}$$

2.  the polynomial $p_{|V|}^{(|E|)}$ is non-zero, and if $G$ is a tree then $p_j^{(i)}$ is non-zero for 