  We devise a framework for proving tight lower bounds under the counting
exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on
Algorithms, 2014). Our framework allows us to convert classical #P-hardness
results for counting problems into tight lower bounds under #ETH, thus ruling
out algorithms with running time $2^{o(n)}$ on graphs with $n$ vertices and
$O(n)$ edges. As exemplary applications of this framework, we obtain tight
lower bounds under #ETH for the evaluation of the zero-one permanent, the
matching polynomial, and the Tutte polynomial on all non-easy points except for
one line. This remaining line was settled very recently by Brand et al. (IPEC
2016).
