Part-products of $S$-restricted integer compositions

Abstract

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda }=({\lambda }_1,{\lambda }_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable $\mathbf{B}(\vec{\lambda })={\prod }_i{\lambda }_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.

18 pages