Applications of probabilistic techniques to study certain combinatorial structures (like integer partitions or compositions) were originated around 1940 by Erd\"os and Lehner and have since served as a quite powerful tool. In this talk I will attempt to present some of a recent progress in that area which has been made possible largely thanks to a "conditioning device" of Fristedt (1993). In particular, I will discuss a joint (with Rod Canfield and Sylvie Corteel) work concerning partitions with non-negative rth difference (current final version is available at http://www.mcs.drexel.edu/~phitczen).