…one flew east, one flew west, One flew over the cuckoo’s nest. —Children’s folk rhyme
The epigraph refers to a children’s rhyme about birds. The verses are taken from a longer children’s counting rhyme, part of which goes like this:
Three geese in a flock. One flew east, one flew west, One flew over the cuckoo's nest. O-U-T spells OUT, Goose swoops down and plucks you out.
One way to think the meaning of the epigraph is that there are two distinct groups presented: the geese that fly east and the geese that fly west. These groups are going in opposite directions, kind of like the patients versus Nurse Ratched and her hospital staff minions. The goose that flies over the cuckoo’s nest would be McMurphy, because he’s the one who ends up crazy (or cuckoo) in the end because of his lobotomy. Notice how one goose also escapes and is plucked from the "cuckoo’s nest," or the asylum. That one goose would be Chief. But Chief didn’t find out how to escape alone; McMurphy played the savior, teaching Chief how to escape from the ward.