Covariance within Random Integer Compositions

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 ${\lim }_{N\to \mathrm{\infty }}\rho (N)$ 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 $\ge 2$.