Bicrucial $k$-power-free permutations

TL;DR

This paper proves the existence of arbitrarily long bicrucial and right-crucial permutations that avoid certain repetitive patterns, advancing understanding of pattern-avoiding permutations.

Contribution

It establishes the existence of arbitrarily long bicrucial and right-crucial $k$-power-free permutations for all $k \,\geq\, 3$, providing new constructions and bounds.

Findings

Existence of arbitrarily long bicrucial $k$-power-free permutations for all $k\geq 3

Existence of right-crucial $k$-power-free permutations of length at least $(k-1)(2k+1)$ for all $k\geq 3

New bounds and constructions for pattern-avoiding permutations

Abstract

In this work, we prove that for every $k\geq 3$ there exist arbitrarily long bicrucial $k$-power-free permutations. We also show that for every $k\geq 3$ there exist right-crucial $k$-power-free permutations of any length at least $(k-1)(2k+1)$.