Permutations all of whose patterns of a given length are distinct

Journal article, 2013

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for every k. We conjecture an even more precise result, based on data for small values of k.