  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.
