On the Multiplicity of Parts in a Random Composition of a Large Integer

In this paper we study the following question posed by H. S. Wilf: what is, asymptotically as n → ∞, the probability that a randomly chosen part size in a random composition of an integer n has multiplicity m? More specifically, given positive integers n and m, suppose that a composition λ of n is selected uniformly at random and then, out of the set of part sizes in λ, a part size j is chosen uniformly at random. Let P(A (m) n ) be the probability that j has multiplicity m. We show that for fixed m, P(A (m) n ) goes to 0 at the rate 1/ lnn. A more careful analysis uncovers an unexpected result: (lnn)P(A (m) n ) does not have a limit but instead oscillates around the value 1/m as n → ∞. This work is a counterpart of a recent paper of Corteel, Pittel, Savage, and Wilf, who studied the same problem in the case of partitions rather than compositions.