Counting King Permutations on the Cylinder

TL;DR

This paper investigates the properties and distribution of cylindrical king permutations, a special class of permutations with specific adjacency constraints, including their asymptotic behavior and a new permutation parameter called cyclic bonds.

Contribution

It introduces the concept of cylindrical king permutations, derives recursions, and analyzes their asymptotic distribution and a new permutation parameter, cyclic bonds.

Findings

Derived recursions for cylindrical king permutations

Calculated their asymptotic proportion among king permutations

Analyzed the distribution of cyclic bonds in permutations

Abstract

We call a permutation $\sigma=[\sigma_1,\dots,\sigma_n] \in S_n$ a {\em cylindrical king permutation} if $ |\sigma_i-\sigma_{i+1}|>1$ for each $1\leq i \leq n-1$ and $|\sigma_1-\sigma_n|>1$. We present some results regarding the distribution of the cylindrical king permutations, including some interesting recursions. We also calculate their asymptotic proportion in the set of the 'king permutations', i.e. the ones which satisfy only the first of the two conditions above. With this aim we define a new parameter on permutations, namely, the number of {\em cyclic bonds} which is a modification of the number of bonds. In addition, we present some results regarding the distribution of this parameter.