Random partitions of integers, bounded both in the number of summands and the size of each summand are considered, subject to the probability measure which assigns a probability proportional to some fixed positive number to the power of the number being partitioned. This corresponds to considering Young diagrams confined to a rectangle. When the rectangle grows, and diagrams are rescaled, the probability measure degenerates to a delta measure on a continuous curve, the limit shape. In the intermediate scaling, the fluctuations around the limit shape turn out to be governed by an Ornstein-Uhlenbeck process. Similar behaviour occurs in the related models bounded only on one side or not at all, which were studied by Vershik and others.