Covariance within Random Integer Compositions

Abstract

Fix a positive integer $N$. Select an additive composition $ξ$ of $N$ uniformly out of $2^{N-1}$ possibilities. The interplay between the number of parts in $ξ$ and the maximum part in $ξ$ is our focus. It is not surprising that correlations $ρ(N)$ between these quantities are negative; we earlier gave inconclusive evidence that $\lim_{N \to \infty} ρ(N)$ 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 $ξ$ is a 1-free composition, i.e., possessing only parts $\geq 2$.