Permutations all of whose patterns of a given length are distinct

For each integer k⩾2k⩾2, let F(k)F(k) denote the largest n   for which there exists a permutation σ∈Snσ∈Sn all of whose patterns of length k   are distinct. We prove that F(k)=k+⌊2k−3⌋+ϵk, where εk∈{−1,0}εk∈{−1,0} for every k. We conjecture an even more precise result, based on data for small values of k.