If a row of people must reorder themselves so that nobody is sitting in the same seat as before, that's a derangement. Is there a name for the derangement wherein no person in their new seat is has the same neighbors as before? I would have called this a double derangement but that refers to a property of two derangements of a given set, the each being a derangement of the other and of the original ordering.
Consider derangements of $123456$:
- $234561$ has almost all the same neighbors (256 of these)
- $635142$ has no element near another (27 of these)
- $351642$ also has no element formerly on the end still at the end (19 of these)
- $314625$ can be arranged in a circle and not have the same neighbors (18 of these)