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.

0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 0, 732, 0, 720, 48, 0, 0

Offset: 1

Author

Hugo Pfoertner, May 03 2006

Keywords

• hard
• more
• nonn

Comments

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

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

No solutions exist if n is a prime power. Proved by W. Edwin Clark, Sep 14 2005

Murray Klamkin proved that solutions do exist if n is not a prime power. - Jonathan Sondow, Oct 17 2013

Examples

			The smallest n for which a solution exists is n=6 with 4 solutions:
...........Weight
......1..2..3..4..5..6
.Count...at.position
..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
Configurations 1 is the mirror image of configuration 4, ditto for configurations 2 and 3. Therefore a(6)=2.
		

Links

• Andrew Bernoff, Bernoff's Puzzler, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005
• Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
• Hugo Pfoertner, Balanced weights on circle (Tables of configurations)
• G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.

Crossrefs

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.

Programs

Mathematica

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 *)

Formula

a(A000961(n)) = 0, a(A024619(n)) > 0. - Jonathan Sondow, Oct 17 2013