Permutations all of whose patterns of a given length are distinct

Abstract

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation σ\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. Suggestions for further investigations along these lines are discussed.

9 pages, no figures. This is Version 2: A small error at the very end of the proof in Section 2 has been corrected