I'm using Guice to handle my dependency injection needs, which is why I had a hard time getting Jackson integrated with my Jersey project. Since Genson wasn't doing what I wanted I decided to try Jackson, again. I tried changing several things around until it just worked, trying different suggestions on SO and Google.
And now the following gives the output as expected in my Sandbox project:
ObjectMapper mapper = new ObjectMapper();
AnnotationIntrospector introspector = new JaxbAnnotationIntrospector();
mapper.getSerializationConfig().setAnnotationIntrospector(introspector);
String jsonData = mapper.writeValueAsString(mc);
System.out.println(jsonData);
{"mapOfSets":{"set1":["1","11"],"set0":["0"],"set2":["2","222","22"]}}