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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)
smichr
  • 560
  • First step, check OEIS: https://oeis.org/A288208. The only reference is this other MSE question: https://math.stackexchange.com/questions/2309850. It is safe to say this concept has not been researched much, so I am pretty sure there is no widely used name. – Mike Earnest Aug 05 '21 at 23:46
  • OK, now I see how useful OEIS can be! Thanks for the reminder @MikeEarnest. – smichr Aug 06 '21 at 15:31

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