Invariant permutations for consecutive k-out-of-n cycles

Consecutive-k-out-of-n cycles are proposed as topologies for k-loop computer networks and describe a circular system of n components where the system fails if and only if any k consecutive components all fail. Suppose that the components are interchangeable. The the question arises as to which permutation maximizes the system reliability, assuming that the components have unequal reliabilities. If there exists on optimal permutation which depends on the ordering, but not the values, of the component reliabilities, then the system (and the permutation) is called invariant. The circular system is found to be not invariant except for k=1, 2, n-2, n-1, and n