There seem to be some screwing conditional on the paintings. Anecdotally, the parasol painting seems to come up a lot less, but that might be just me reading into the fact that it's described as hidden in the corner. I do recall on halloween it taking 40+ energy for it to come up though. I've had it come up twice in ~4 turns though, so probably not a minimum wait between it, more likely just a flat rejection rate.
So I started looking into it and collecting some data. With the following bonuses (neither of which seem like they should apply)
Greatly [6] Increased Chance of Combat Encounters
Highly [4] Increased Chance of PvP Encounters
I got the following encounters:
Nothing Too Dangerous | 35 | 4x |
Starry Sky | 10 | 1x |
Parasol Portrait | 6 | 1x |
Recommended Donation | 37 | 4x |
Large Landscape | 20 | 2x |
Sunset Painting | 7 | 1x |
Suckling Pig | 9 | 1x |
The choice encounters definitely seem to come up a lot more often than the others. It's hard to tell if Emily's painting comes up less often than most of the other paintings with this limited data.