The maximal length of a \(k\)-separator permutation (Q405306)

Summary: A permutation \(\sigma\in S_n\) is a \(k\)-separator if all of its patterns of length \(k\) are distinct. Let \(F(k)\) denote the maximal length of a \(k\)-separator. \textit{P. Hegarty} [``Permutations all of whose patterns of a given length are distinct'', J. Comb. Theory, Ser. A 120, No. 7, 1663--1671 (2013; \url{doi:10.1016/j.jcta.2013.06.006})] showed that \(k+\left\lfloor\sqrt{2k-1}\right\rfloor-1\leq F(k)\leq k+\left\lfloor\sqrt{2k-3}\right\rfloor\), and conjectured that \(F(k)=k+\left\lfloor\sqrt{2k-1}\right\rfloor-1\). This paper will strengthen the upper bound to prove the conjecture for all sufficiently large \(k\) (in particular, for all \(k\geq 320801\)).