Thu, 29 Nov 2018
Take !!N!! equally-spaced points on a circle. Now connect them in a loop: each point should be connected to exactly two others, and each point should be reachable from the others. How many geometrically distinct shapes can be formed?
For example, when !!N=5!!, these four shapes can be formed:
(I phrased this like a geometry problem, but it should be clear it's actually a combinatorics problem. But it's much easier to express as a geometry problem; to talk about the combinatorics I have to ask you to consider a permutation !!P!! where !!P(i±1)≠P(i)±1!! blah blah blah…)
For !!N<5!! it's easy. When !!N=3!! it is always a triangle. When !!N=4!! there are only two shapes: a square and a bow tie.
But for !!N=6!!, I found it hard to enumerate. I think there are nine shapes but I might have missed one, because I know I kept making mistakes in the enumeration, and I am not sure they are all corrected:
It seems like it ought not to be hard to generate and count these, but so far I haven't gotten a good handle on it. I produced the !!N=6!! display above by considering the compositions of the number !!6!!:
(Actually it's the compositions, modulo bracelet symmetries — that is, modulo the action of the dihedral group.)
But this is fraught with opportunities for mistakes in both directions. I would certainly not trust myself to hand-enumerate the !!N=7!! shapes.
[ Addendum: For !!N=6!! there are 12 figures, not 9. For !!N=7!!, there are 39. Further details. ]