So, a student asked me about a homework question that is a sub-problem of the structure generation problem. Basically, it was to count the number of chemical structures with exactly one cycle given the elemental formula. Of course, the best solution here is probably to use the PolyÃ¡ Enumeration Theorem since all that was asked for was a count (enumeration) of the structures.

Naturally, I have a different way to do this - especially since I don't really understand the mathematics of PET enough to implement it. So:

The image shows a rough overview of how I might list all of the structures with a single cycle. It takes a number of necklaces (one shown), and a number of trees, and glues the one to the other in all possible ways. The word 'necklace' here is specifically the combinatorial object; so the cyclic sequence CNCNO is the same as CNOCN since you can rotate one to get the other.

One tricky decision here is whether to add multiple bonds to the necklace before or after a…

Naturally, I have a different way to do this - especially since I don't really understand the mathematics of PET enough to implement it. So:

The image shows a rough overview of how I might list all of the structures with a single cycle. It takes a number of necklaces (one shown), and a number of trees, and glues the one to the other in all possible ways. The word 'necklace' here is specifically the combinatorial object; so the cyclic sequence CNCNO is the same as CNOCN since you can rotate one to get the other.

One tricky decision here is whether to add multiple bonds to the necklace before or after a…