Permutations all of whose patterns of a given length are distinct

TL;DR

This paper determines the maximum length of permutations where all patterns of a fixed size are distinct, providing an exact formula involving the integer k and its square root.

Contribution

It establishes a precise formula for F(k), the largest permutation length with all distinct k-patterns, advancing understanding of pattern uniqueness in permutations.

Findings

F(k) = k + loor{\sqrt{2k-3}} + e_k

e_k ext{ is either -1 or 0 for all } k

Provides a foundation for further exploration of pattern uniqueness

Abstract

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. Suggestions for further investigations along these lines are discussed.