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 $n\rightarrow \infty$, 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 $\lambda$ of n is selected uniformly at random and then, out of the set of part sizes in $\lambda$, a part size j is chosen uniformly at random. Let $\P(A_n^{(m)})$ be the probability that j has multiplicity m. We show that for fixed m, $\P(A_n^{(m)}$) goes to 0 at the rate $1/\ln n$. A more careful analysis uncovers an unexpected result: $(\ln n)\P(A_n^{(m)})$ does not have a limit but instead oscillates around the value $1/m$ as $n\to\infty$. 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.