A118888 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk such that the remaining imbalance is minimized.

1, 1, 1, 1, 1, 2, 1, 2, 3, 24, 1, 732, 1, 720, 48, 144, 2

Offset: 1

Author

Hugo Pfoertner, May 26 2006

Keywords

• hard
• more
• nonn

Comments

The position of weight 1 is kept fixed at position 1. Mirror configurations are counted only once. For n not a prime power, the sequence equals A118887.

Examples

			a(5)=1: The configuration minimizing the remaining imbalance with respect to the center of the circle is [1 4 3 2 5] (and its mirror image).
Examples of minimum imbalance configurations not in A118887:
a(7)=1: [1 4 7 2 3 5 6];
a(8)=2: [1 4 7 3 6 2 5 8], [1 7 4 3 6 5 2 8];
a(9)=3: [1 5 9 2 7 3 4 8 6], [1 5 9 4 2 6 7 3 8], [1 6 5 4 9 2 3 7 8];
a(11)=1: [1 8 9 5 2 6 10 7 3 4 11];
a(13)=1: [1 2 7 12 13 4 5 3 8 6 11 9 10];
a(16)=144: lexicographically earliest [1 3 5 13 16 7 10 2 14 4 6 9 12 8 11 15];
a(17)=2: [1 7 3 17 10 9 15 2 14 6 5 4 16 8 13 12 11],
[1 8 9 3 16 4 12 13 14 2 10 5 6 7 17 11 15] and their mirror configurations (e.g. [1 11 12 13 8 ...]) both produce a center of gravity with distance 2.1884*10^(-7) from the center of a circle with radius 1. All other configurations produce greater distances, e.g. [1 3 11 16 9 5 7 12 14 4 10 8 2 15 13 6 17] -> 2.5126*10^(-7). - _Hugo Pfoertner_, Oct 24 2019
		

Links

• G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.

Crossrefs

Cf. A118887, A326921.

Extensions

a(17) corrected by Hugo Pfoertner, Oct 24 2019