Covariance within Random Integer Compositions

TL;DR

This paper investigates the statistical relationship between the number of parts and the maximum part size in random integer compositions, revealing that their correlation tends to zero, contrary to previous conjectures of negative dependence.

Contribution

The study provides new insights into the asymptotic dependence structure of parts in random compositions, challenging earlier assumptions about negative correlation.

Findings

Correlation between parts and maximum part tends to zero as N increases

Negative correlation observed for finite N, but asymptotic independence is suggested

Results extend to 1-free compositions with parts ≥ 2

Abstract

Fix a positive integer $N$. Select an additive composition $\xi$ of $N$ uniformly out of $2^{N-1}$ possibilities. The interplay between the number of parts in $\xi$ and the maximum part in $\xi$ is our focus. It is not surprising that correlations $\rho(N)$ between these quantities are negative; we earlier gave inconclusive evidence that $\lim_{N \to \infty} \rho(N)$ is strictly less than zero. A proof of this result would imply asymptotic dependence. We now retract our presumption in such an unforeseen outcome. Similar experimental findings apply when $\xi$ is a 1-free composition, i.e., possessing only parts $\geq 2$.