Asymptotic normality of integer compositions inside a rectangle

TL;DR

This paper demonstrates that the distribution of integer compositions constrained within a rectangle approaches a normal distribution as the dimensions grow large, with the mean roughly at half the rectangle's area.

Contribution

It establishes the asymptotic normality of the distribution of restricted integer compositions within a rectangle, providing a probabilistic understanding of their behavior.

Findings

Distribution is approximately normal for large dimensions

Mean value of the composition is roughly half the rectangle's area

Results apply to compositions with at most m parts and size at most l

Abstract

Among all restricted integer compositions with at most $m$ parts, each of which has size at most $l$, choose one uniformly at random. Which integer does this composition represent? In the current note, we show that underlying distribution is, for large $m$ and $l$, approximately normal with mean value $\frac{ml}{2}$.