  In this paper we explore a relationship between the topology of the complex
hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the
combinatorics of certain spaces of degree-$n$ polynomials over a finite field
$\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace
formula and work of Lehrer and Kim. We use it to prove a correspondence between
a representation-theoretic convergence result on the cohomology algebras
$H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability
result for certain polynomial statistics on monic squarefree polynomials over
$\mathbb{F}_q$ with nonzero constant term. This result is the type B/C analogue
of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new
proof of their theorem. To establish these convergence results, we realize the
sequences of cohomology algebras of the hyperplane complements as
FI$_\mathcal{W}$-algebras finitely generated in FI$_\mathcal{W}$- degree $2$,
and we investigate the asymptotic behaviour of general families of algebras
with this structure. We prove a negative result implying that this structure
alone is not sufficient to prove the necessary convergence conditions. Our
proof of convergence for the cohomology algebras involves the combinatorics of
their relators.
