On $P$-crucial square-free permutations

TL;DR

This paper proves the existence of specific $P$-crucial square-free permutations for an infinite family of lengths, extending previous finite results and deepening understanding of permutation structures avoiding squares.

Contribution

It establishes the existence of $oxed{0,1,8m+4,8m+5}$-crucial square-free permutations for all integers $m \, \geq \, 2$, generalizing prior finite cases.

Findings

Existence of $oxed{0,1,8m+4,8m+5}$-crucial permutations for all $m \geq 2$

Construction method for these permutations

Extension of previous results to infinite family of lengths

Abstract

A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A square-free permutation of length $n$ is $P$-crucial, where $P$ is a subset of $\{0,1,\ldots,n\}$, if any of its extensions in any position from the set $P$ contains a square. In 2015, Gent, Kitaev, Konovalov, Linton and Nightingale initiated the study of $P$-crucial square-free permutations. In particular, they showed that $\{0,1,n-1,n\}$-crucial square-free permutations of length $n$, where $n\leq 22$, exist if and only if $n=17$ or $n=21$. In this work, we prove that for any $m\geq 2$ there exists a $\{0,1,8m+4,8m+5\}$-crucial square-free permutation of length $8m+5$.