New Year's Countdown Day 8: Eight Seated People

Eight people are sitting around a circular table. In how many ways can they reseat themselves so that nobody is seated to the immediate right of the same person as before?

As an explicit example, if they were originally seated as follows: \[ \quad \, \, \ \text{A} \, \, \quad \\ \quad \text{H} \quad \, \, \ \ \ \ \text{B} \quad \\ \text{G} \quad \quad \quad \, \, \ \text{C} \\ \quad \text{F} \quad \, \, \ \ \ \ \text{D} \quad \\ \quad \, \, \ \text{E} \, \, \quad \\ \] then they can reseat themselves as follows, and no one would be sitting to the right of the same person as before: \[ \quad \, \, \ \text{A} \, \, \quad \\ \quad \text{B} \quad \, \, \ \ \ \ \text{H} \quad \\ \text{C} \quad \quad \quad \, \, \ \text{G} \\ \quad \text{D} \quad \, \, \ \ \ \ \text{F} \quad \\ \quad \, \, \ \text{E} \, \, \quad \\ \] Details and Assumptions:

  • Arrangements that are rotations of each other are considered the same.
  • Arrangements that are reflections of each other are considered distinct.
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