In the first step there are two arrows, because the rule (which is the identity) can be applied in two ways to the initial state.I don't understand why there are four arrows for every subsequent step.
Interesting. Have you got any idea why does it behave like that?
All know is that where your(2X2) map had 2,4 then 4 paths, my (3X3) map had 6, 36 then 36.
I suspect a (4X4) map will have 4,64 then 64 paths.