The lengths for which bicrucial square-free permutations exist

TL;DR

This paper classifies all lengths for which bicrucial square-free permutations exist, providing a complete understanding of their possible sizes and structural properties.

Contribution

It offers a complete classification of lengths for which bicrucial square-free permutations exist, advancing the understanding of their combinatorial structure.

Findings

Bicrucial square-free permutations exist for specific lengths.

Complete characterization of lengths with such permutations.

Insights into the structural constraints of square-free permutations.

Abstract

A square is a factor $S = (S_1; S_2)$ where $S_1$ and $S_2$ have the same pattern, and a permutation is said to be square-free if it contains no non-trivial squares. The permutation is further said to be bicrucial if every extension to the left or right contains a square. We completely classify for which $n$ there exists a bicrucial square-free permutation of length $n$.