Place potatoes in a large pot; cover with water by 1 inch, and add 1 tablespoon salt. Bring to a boil; reduce to a simmer, and cook until potatoes are tender, 15 to 20 minutes. With a wire skimmer or slotted spoon, transfer potatoes to a large bowl (reserve water in pot).
To the simmering water, add green beans, and cook until crisp-tender, 2 to 3 minutes. Drain and add to potatoes, along with mayonnaise, vinegar, scallion, and carrot. Toss to combine, and lightly mash; season with salt and pepper, and serve warm.