Buy potatoes of a uniform size so they cook evenly. You can use an egg-free soy-based or other mayonnaise.
Place potatoes in large pot, and cover with water. Bring to a boil, and cook until tender, about 20 minutes. Remove from heat, and, while still warm, cut in quarters. Slice scallions, and toss with potatoes.
Mix remaining ingredients in separate bowl, and toss with warm potatoes. Cover, and chill overnight.