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We consider two families of algebraic varieties $Y_n$ indexed by naturalnumbers $n$: the configuration space of unordered $n$-tuples of distinct pointson $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independentlines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual$S_n$-representations given by a character polynomial, we compute $H^i(Y_n;W_n)$ for all $i$ and all $n$ in terms of double generating functions. Oneconsequence of the computation is a new recurrence phenomenon: the stabletwisted Betti numbers $\lim_{n\to\infty}\dim H^i(Y_n; W_n)$ are linearlyrecurrent in $i$. Our method is to compute twisted point-counts on the$F_q$-points of certain algebraic varieties, and then pass through theGrothendieck-Lefschetz fixed point formula to prove results in topology. Wealso generalize a result of Church-Ellenberg-Farb about the configurationspaces of the affine line to those of a general smooth variety.

Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting