Under ETH, no f(k) n^{o(k/log k)}-time algorithm can approximate k-permutation pattern counts within n^{(1/2-ε)k} factor, matching exact-counting hardness.
[MT04] Adam Marcus and G´ abor Tardos
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Inapproximability of Counting Permutation Patterns
Under ETH, no f(k) n^{o(k/log k)}-time algorithm can approximate k-permutation pattern counts within n^{(1/2-ε)k} factor, matching exact-counting hardness.