Quite impressive. It does not come near to being what I had hoped for but that is my fault. I was so close to the problem that I could not see the forest through the trees. I naively assumed that counting the ways would mean generating them and as you and others have shown - this is a very poor assumption.
I am very happy that you have contributed though because not only do I know a little bit more about math theory, I have a new way of looking at problems.
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