On the poset of King-Non-Attacking permutations

TL;DR

This paper explores the structure and properties of the poset formed by king-non-attacking permutations under containment, including insights into its M"obius function.

Contribution

It introduces the poset of king-non-attacking permutations, analyzing its structure and M"obius function, which is a novel approach in permutation pattern research.

Findings

Characterization of the poset structure

Results on the M"obius function of the poset

Insights into permutation containment relations

Abstract

A king-non-attacking permutation is a permutation $\pi \in S_n$ such that $|\pi(i)-\pi(i-1)|\neq 1$ for each $i \in \{2,\dots,n\}$. We investigate the structure of the poset of these permutations under the containment relation, and also provide some results on its M\"obius function.