Covariance within Random Integer Compositions

10/13/2020

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by   Steven Finch, et al.

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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 →∞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 ≥ 2.