cp's OEIS Frontend

A118887 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.

%I A118887 #32 Oct 22 2019 16:26:21
%S A118887 0,0,0,0,0,2,0,0,0,24,0,732,0,720,48,0,0
%N A118887 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.
%C A118887 The position of weight 1 is kept fixed at position 1. Mirror configurations are counted only once. Proposed in the seqfan mailing list by _Brendan McKay_, Sep 12 2005
%C A118887 Also number of permutations p1,p2,...,pn such that the polynomial p1 + p2*x + ... + pn*x^(n-1) has exp(2*pi*i/n) as a zero. Also number of equiangular polygons whose sides are some permutation of 1,2,3,...,n. - _T. D. Noe_, Sep 13 2005
%C A118887 No solutions exist if n is a prime power. Proved by _W. Edwin Clark_, Sep 14 2005
%C A118887 Murray Klamkin proved that solutions do exist if n is not a prime power. - _Jonathan Sondow_, Oct 17 2013
%H A118887 Andrew Bernoff, <a href="http://www.math.hmc.edu/muddmath/v4n1.pdf">Bernoff's Puzzler</a>, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005
%H A118887 Marius Munteanu and Laura Munteanu, <a href="http://dx.doi.org/10.4236/am.2013.410197">Rational equiangular polygons</a>, Applied Math., 4 (2013), 1460-1465.
%H A118887 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a118888.zip">Balanced weights on circle</a> (Tables of configurations)
%H A118887 G. J. Woeginger, <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.09.849">Nothing new about equiangular polygons</a>, Amer. Math. Monthly, 120 (2013), 849-850.
%F A118887 a(A000961(n)) = 0, a(A024619(n)) > 0. - _Jonathan Sondow_, Oct 17 2013
%e A118887 The smallest n for which a solution exists is n=6 with 4 solutions:
%e A118887 ...........Weight
%e A118887 ......1..2..3..4..5..6
%e A118887 .Count...at.position
%e A118887 ..1...1..4..5..2..3..6
%e A118887 ..2...1..5..3..4..2..6
%e A118887 ..3...1..6..2..4..3..5
%e A118887 ..4...1..6..3..2..5..4
%e A118887 Configurations 1 is the mirror image of configuration 4, ditto for configurations 2 and 3. Therefore a(6)=2.
%t A118887 Needs["DiscreteMath`Combinatorica`"]; Table[eLst=E^(2.*Pi*I*Range[n]/n); Count[(Permutations[Range[n]]), q_List/;Chop[q.eLst]===0]/(2n), {n,10}] (* very slow for n>10 *) (* _T. D. Noe_, May 05 2006 *)
%Y A118887 Cf. A118888 (configurations with minimum imbalance), A063697 (positions of positive coefficients in cyclotomic polynomial in binary), A063699 (positions of negative coefficients in cyclotomic polynomial in binary), A326921.
%K A118887 hard,more,nonn
%O A118887 1,6
%A A118887 _Hugo Pfoertner_, May 03 2006