Part-products of $S$-restricted integer compositions

TL;DR

This paper studies the properties of products of parts in $S$-restricted compositions of integers, showing that for uniform random compositions, the product is asymptotically lognormal, using a novel combinatorial decomposition technique.

Contribution

It introduces a new combinatorial method to analyze $S$-restricted compositions and proves the asymptotic lognormality of the product of parts in uniform random compositions.

Findings

The product of parts in uniform $S$-restricted compositions is asymptotically lognormal.

A new combinatorial decomposition technique is developed for analyzing compositions.

The results extend understanding of the probabilistic structure of restricted 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 ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.