4 research outputs found
The maximal length of a k-separator permutation
A permutation Ο β 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 + ββ2k β 1β β 1 β€ F(k) β€ k + ββ2k β 3β, and conjectured that F(k) = k + ββ2k β 1β β 1. This paper will strengthen the upper bound to prove the conjecture for all sufficiently large k (in particular, for all k β₯ 320801).United States. Dept. of Energy. Division of Materials Sciences and Engineering (Grant 1062709)United States. National Security Agency (Grant H98230-11-1-0224
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 \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
Prolific permutations and permuted packings : downsets containing many large patterns
A permutation of n letters is k-prolific if each (n - k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations of m letters exist for every m >= k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds