<s>[INST] In how many ways can you arrange a circle of partners so that no partners are touching? There are a lot of similar questions to this but none that is quite the same so I figured I would ask a new question. The problem is you have a group of people that came in pairs, in how many ways can the $N$ people be arranged in a circle such that no person in the circle is holding hands with the partner they came with.
I have worked out by hand that for two couples(four people) the answer is $2$ ways
For three couples ($6$ people) the answer is $32$ ways
Also it should be noted that a rotated circle is not counted as a different arrangement. 
If the answer posted below is correct the OEIS sequence is http://oeis.org/A129348 Can any one else confirm that this is correct?
 [/INST] Here is the calculation for $8:$  
Seat one person of the first couple.  This fixes the rotation of the circle.  Seat the second person of the first couple.  The remaining seats can be in groups of $1+5 (2$ ways), $2+4 (2$ ways), or $3+3, (1$ way).  
From $1+5$ we can go to $4$ in $4$ ways, $1+3$ in $6$ ways, $2+2$ in $2$ ways, $1+1+2$ in $4$ ways, or $1+1+1+1$ in $2$ ways.
Similarly we can go from $2+4$ to  $1+3$ in $8$ ways or to $1+1+2$ in $12$ ways.
We can go from $3+3$ to $1+3$ in $4$ ways, $2+2$ in $8$ ways, $1+1+2$ in $8$ ways, or $1+1+1+1$ in $2$ ways.  
Multiplying out, after two couples we can have $4$ in $8$ ways, $1+3$ in $32$ ways, $2+2$ in $12$ ways, $1+1+2$ in $40$ ways or $1+1+1+1$ in $6$ ways.
Finally, from $4$ we can place the last two couples in $8$ ways, $1+3$ in $8$ ways, $2+2$ in $16$ ways, $1+1+2$ in $16$ ways, and $1+1+1+1$ in $24$ ways.
The total comes out $1296$
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