Thu, 29 Nov 2018
How many kinds of polygonal loops? (part 2)
And I said I thought there were nine analogous figures with six points.
Rahul Narain referred me to a recent discussion of almost this exact question on Math Stackexchange. (Note that the discussion there considers two figures different if they are reflections of one another; I consider them the same.) The answer turns out to be OEIS A000940. I had said:
I missed three. The nine I got were:
And the three I missed are:
I had tried to break them down by the arrangement of the outside ring of edges, which can be described by a composition. The first two of these have type !!1+1+1+1+2!! (which I missed completely; I thought there were none of this type) and the other has type !!1+2+1+2!!, the same as the !!015342!! one in the lower right of the previous diagram.
I had ended by saying:
Good call, Mr. Dominus! I considered filing this under “oops” but I decided that although I had gotten the wrong answer, my confidence in it had been adequately low. On one level it was a mistake, but on a higher and more important level, I did better.
I am going to try the (Cauchy-Frobenius-)Burnside-(Redfield-)Pólya lemma on it next and see if I can get the right answer.
Thanks again to Rahul Narain for bringing this to my attention.
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