Given an n-dimensional electrically neutral, solid metal ball (a point for n=0; a rod, n=1; a disc, n=2; a solid ball, n=3; ...), place N=(n+1)! identical ions on the ball. As one of my favorite physics professors used to say, forget the mathematics, intuitively I expect the ions to equilibrate to the vertices of an n-D [permutohedron][1] inscribed in the n-ball (e.g., a hexagon inscribed in the circumference of a disc with 6 ions and a [truncated octahedron][2] for n=3 ). Similarly with N= [2(n+1)]! /[(n+1)! (n+2)!] (the [Catalan numbers][3]), I expect to see an [associahedron][4] (a Stasheff polytope, e.g., a pentagon for the disc with 5 ions).

Is my intuition correct? Has anyone seen this worked out mathematically, as an extremum problem? An electrostatics simulation for n=3 would be interesting also. (See OEIS [A019538][5] for refined f-vectors of permutohedra / permutahedra and [A133437][6] for associahedra.)


  [1]: http://en.wikipedia.org/wiki/Permutohedron
  [2]: http://en.wikipedia.org/wiki/Truncated_octahedron
  [3]: http://oeis.org/A000108
  [4]: http://en.wikipedia.org/wiki/Associahedron
  [5]: https://oeis.org/A049019
  [6]: http://oeis.org/A133437