The bottom row is close to what I imagine, but without IO ports on the same edge being allowed to connect to each other (though that is also an interesting problem). These would be the three diagrams for the square:
The middle one makes a single loop which is one-third of them, and n=4/2=2 in this case. My guess for how to prove the recurrence is to “glue” polygons together:
There are n+1 pairs of sizes (k,n+1−k) we can glue together (if you’re okay with 2-sided polygons), but I haven’t made much progress in this direction. All I’ve found is gluing two polygons together decreases the number of loops by zero, one or two.
The bottom row is close to what I imagine, but without IO ports on the same edge being allowed to connect to each other (though that is also an interesting problem). These would be the three diagrams for the square:
The middle one makes a single loop which is one-third of them, and n=4/2=2 in this case. My guess for how to prove the recurrence is to “glue” polygons together:
There are n+1 pairs of sizes (k,n+1−k) we can glue together (if you’re okay with 2-sided polygons), but I haven’t made much progress in this direction. All I’ve found is gluing two polygons together decreases the number of loops by zero, one or two.