A Method and Two Algorithms on the Theory of Partitions

In this note we describe a general principle for selecting at random from a collection of combinatorial objects, where “at random” means in such a way that each of the objects has equal probability, a priori, of being selected. We apply this principle by displaying two algorithms, the first of which will select a random partition of an integer n, and the second will choose a random partition of a set S of n elements. We begin by stating the general principle, albeit rather vaguely. Let a, be the number of combinatorial objects of order n. Suppose the numbers a, satisfy a recurrence relation of the form