Permutations all of whose patterns of a given length are distinct

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