The Universe of Discourse

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:

$Four displays, each with five dark gray dots arranged at the vertices of a regular pentagon. In each display the dots are connected with purple lines, but each in a different order. If the dots were numbered 0-1-2-3-4 in clockwise order, the four figures have purple lines connecting them, respectively, in the orders 0-1-2-3-4-0, 0-1-3-2-4-0, 0-2-1-4-3-0, and 0-2-4-1-3-0. The first of these is a plain pentagon, and the last is a five-pointed star. The middle two are less symmetric.$

(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:

$This time there are nine displays, each with six dots. The connections are, respectively, 012345 (a hexagon), 015432, 032145 (two diamonds), 015234 (highly irregular), 014523 (three triangles that share a vertex in the center), 013254, 023154, 031254, and 035142.$

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!!:

Composition	How many loops?
6	1
1+5	—
2+4	1
3+3	1
1+1+4	—
1+2+3	1
2+2+2	2
1+1+1+3	—
1+1+2+2	1
1+2+1+1	1
1+1+1+1+2	—
1+1+1+1+1+1	1
Total	9 (?)

(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. ]

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