Okay, "open rotor" I understand. But "ducted open rotor", what's with that? Better ask Boeing. Its name is on the recent patent application my colleague Steve Trimble came across when browsing the US patent office website. And it actually is for an open rotor with a duct around it....
The patent application talks about "an aerodynamically shaped propeller duct that...fully houses a propeller driven by an engine...[and] significantly reduces the noise associated with a turboprop engine". But the applicability of the design to open rotors, unducted fans, propfans or whatever you want to call them is clear. Is this a sign of Boeing's thinking on how it could install open rotors on its next-generation single aisle?
Noise is the big issue with open rotors, and fitting a duct around them would be one answer, although the drag and weight of a duct that large would seem to be issues. But if the fuel and emissions savings from open rotors are substantial enough, maybe those are penalties worth paying.