  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 $4$.
