What do you mean by non-compatible? If you are referring to 'companion planting' don't worry about it. It's nonsense. The only criterion you need be concerned regarding distance is how big they'll get and when they're planted. There's nothing about those crops which says they can't be planted together other than their size, light requirements and growing season. Without knowing your location we can't tell when you should be planting these crops. In my climate sprouts are a winter veg, planted in June/July, and would not be taking up much space when corn and beans are in the ground. Corn needs to be planted in blocks of a good number to ensure pollination. You don't give the width of the bed but six feet sounds too small a space for a good block of corn plus pole beans.

The 4 ×22 ..ft 15 ''

Concur: onions and Brussel Sprouts cool season crops. pole beans and corn are are hot weather crops. Their growing seasons have minimum overlap except in long term moderate climates which don't have normal seasonal temperature variations.