Suggestion for a sequence: weights on a circle

[Hugo Pfoertner]

Brendan McKay wrote:
> 
> A recent puzzle in New Scientist suggests the following.
> 
> Suppose you have n objects, with weights 1,2,3,...,n.
> (That is, exactly one object with each of those weights.)
> 
> How many ways are there to place these objects evenly spaced
> around the circumference of a disk so that the disk will
> exactly balance on the centre point?
> 
> In algebraic terms, how many permutations p1,p2,...,pn
> are such that the polynomial p1 + p2*x + ... + pn*x^(n-1)
> has exp(2 pi i/n) as a zero?
> 
> An example for n=10 is [2, 9, 1, 8, 5, 7, 4, 6, 3, 10]. I don't even
> know which n have a solution. Certainly not n=1,2,3,4. For n=10
> there are 480 solutions, or 48 equivalence classes if "1" is placed
> in a fixed position, or 24 equivalence classes if the mirror
> image is regarded as the same.
> 
> Brendan.

I've written a program
http://www.randomwalk.de/sequences/balcir.for
(calling my lexicographic permutation function
http://www.randomwalk.de/sequences/lpg.txt)
to get the first few terms of this sequence. I keep the mass 1 fixed at
(1,0) on the unit circle.
The smallest n for which a solution exists is n=6 with 4 solutions
  Count       Mass
        1  2  3  4  5  6
        at  positions
     1  1  4  5  2  3  6
     2  1  5  3  4  2  6
     3  1  6  2  4  3  5
     4  1  6  3  2  5  4

No solutions for n=7,8,9,

48 solutions (as indicated above) for n=10. The lexicographically first
solution is:
 [1  4  5  8  9  2  3  6  7 10]
the others can be found at
http://www.randomwalk.de/sequences/balcir10.txt

No solution for n=11.

1464 solutions for n=12. Lexicographically first:
[1  2  6 10  8  9  4  5  3  7 11 12]
List: http://www.randomwalk.de/sequences/balcir12.txt

No solution for n=13.

1440 solutions for n=14. Lexicographically first:
[1  4  5  8  9 12 13  2  3  6  7 10 11 14]
List: http://www.randomwalk.de/sequences/balcir14.txt

96 solutions for n=15. Lexicographically first:
[1  5  9 10 14  3  4  8 12 13  2  6  7 11 15]
List: http://www.randomwalk.de/sequences/balcir15.txt

n=16 seems doable, but will take a few days CPU.

The suggested sequence (number of ways to arrange the weights) would
start (first term n=1)

0 0 0 0 0 4 0 0 0 48 0 1464 0 1440 96

Hugo Pfoertner