The poset of king permutations on a cylinder

TL;DR

This paper studies the structure of cylindrical king permutations, a special class of permutations modeling non-attacking kings on a cylindrical chessboard, and investigates their poset properties, including maximal gaps and prolific permutations.

Contribution

It introduces the poset of cylindrical king permutations, analyzes its structure, and establishes bounds on gaps and criteria for prolific permutations.

Findings

Maximal gap between permutations is 4.

Characterization of k-prolific cylindrical king permutations.

Structural insights into the poset of cylindrical king permutations.

Abstract

A permutation $\sigma=[\sigma_1,\dots,\sigma_n] \in S_n$ is called a {\em cylindrical king permutation} if $ |\sigma_{i+1}-\sigma_{i}|>1$ for each $1\leq i \leq n-1$ and $|\sigma_1-\sigma_n|>1$. The name comes from the the way one can see these permutations as describing locations of $n$ kings on a chessboard of order $n\times n$ in such a way that (each row and each column contains exactly one king and) no two kings are attacking each other, with the additional condition that a king can move off a certain row and reappear at the beginning of that row. In a recent paper, we dealt with the more general set of 'king permutations' i.e. the ones which satisfy only the first of the two conditions above. This set constitutes a poest under the well known containment relation on permutations. In this article we investigate the sub-poset of the cylindrical king permutations and its structure. We examine those cylindrical king permutations whose downset is as large as possible in the upper ranks. We use a modification of Manhattan distance of the plot of a permutation and some of its applications to the cylindrical context to find a criterion for such a permutation to be $k-$ prolific. One of our main results is that the maximal gap between two permutations in the poset of cylindrical permutations is $4$.