The maximal length of a k-separator permutation

Abstract

A permutation σ ∈ S[subscript 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. Hegarty (2013) showed that k + ⌊√2k − 1⌋ − 1 ≤ F(k) ≤ k + ⌊√2k − 3⌋, and conjectured that F(k) = k + ⌊√2k − 1⌋ − 1. This paper will strengthen the upper bound to prove the conjecture for all sufficiently large k (in particular, for all k ≥ 320801).