In how many ways can $x$ people be seated at a round table so that all will not have the same neighbours in any two arrangements?
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1Are we talking about the number of permutations the second time at the table? – Joffan Jan 17 '15 at 08:29
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OK, here's the question as I understand it.
A group of $X$ friends went for dinner and sat at a round table. A week later, they all went for dinner again and once again sat at a round table. How many possible seating arrangements for the second week allow every participant to have different neighbours from the first week?
For the time being, I'll assume that the youngest participant gets a designated seat, so that rotations can be avoided. On the first visit the seating was abcdefg...
- $X=2, 3, 4$: possible seating arrangements $P=0$.
- $X=5, P=2$: adbec and acebd (reflections)
- $X=6, P=6$: acebfd, acfdbe, adbfce + reflections
- $X=7, P=46$
- $X=8, P=354$
The sequence is at OEIS A078603
Regarding rotations of an arrangement as distinct (and I find facing the bar or facing the window different, personally), the numbers should be multiplied by the number of people in each case.
Joffan
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