the author suggest that the polynomials $p_j^{(i)}$ should be non-zero for all graphs $G$, and all $0 \leq i \leq |E|$ with $j \leq \delta_i$.\
If we restrict ourselves to rational homology, the groups $H_i(\operatorname{Conf}_{\bullet}^{\operatorname{sink}}(G,V);{\mathbb{Q}})$ form a collection of complex symmetric group representations. As the name representation stability suggests, we should be able to say something about these representations as $n$ is allowed to vary. To this end we have the following.\

\[sinkrepstab\] Assume that $|V| = d$. For any $i$, The complex ${\mathfrak{S}}_n$-representations $H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V);{\mathbb{Q}})$ enjoy the following properties:

1.  for any partition $\lambda$, and any integers $n_1 \geq \ldots \geq n_d \geq |\lambda| + \lambda_1$, let $c_{\lambda,n_1,\ldots,n_d}$ be the multiplicity of $S(\lambda)_{n_1,\ldots,n_d}$ in $H_i(\operatorname{Conf}_{\sum_i n_i - (d-1)|\lambda|}^{\operatorname{sink}}(G,V);{\mathbb{Q}})$ (see Definition \[padpar\]). Then the quantity $c_{\lambda,n_1+l,\ldots,n_d+l}$ is independent of $l$ for $l \gg 0$;

2.  if $c_{\lambda,n}$ is the multiplicity of $S(\lambda)_n$ in $H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$, then there exists a polynomial $p(x) \in {\mathbb{Q}}[x]$ of degree $\leq d-1$ such that for all $n \gg 0$, $c_{\lambda,n} = p(n)$;

3.  there is a finite integer $b_i^{(G,V,E)}$, independent of $n$, such that for any irreducible representation $S^{\lambda}$ appearing in $H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$, $\lambda$ has at most $b_i^{(G,V,E)}$ boxes below its $d$-th row.\

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The 