A Climber's Guide to the Teton Range gives us this photo. Buckingham Ridge is labeled Route B, and is referred to as the Southeast Ridge. Route C is the Ellingwood Couloir, and Route A is the South Couloir.

Note that the route cuts way right above the initial 5th class rock to join the central part of the ridge on its easier righthand side.

My partner and I, instead, climbed a line that cut up through the central face, and found it to be doable at about 5.8 - 5.9.