Mono in the partial ordering of measures of disorder

In the previous article1, I developed a new measure of disorder called \(\mathit{Reve}\) to make it easier to compare the values of \(\mathit{Amp}\)2 and \(\mathit{Mono}\). I proved that the new measure was equivalent to \(\mathit{Mono}\), then conjectured that its simpler mathematical defnition would allow to more easily analyze relationships of \(\mathit{Mono}\) with other measures of disorder. I wasn’t exactly wrong about the “analyze relationships of \(\mathit{Mono}\)” part: as a matter of fact it’s what this very blog post is about. \(\mathit{Reve}\), however, plays no part in those newfound results. Instead I’m going to rely on the tried-and-true mathematical trick of re-reading the literature and putting the pieces together. In the following sections, we revisit the measures of disorder \(\mathit{Runs}\)3 and \(\mathit{Mono}\). In that process, we use variants of the classic mergesort algorithm to show through concrete examples how measures of disorder relate to adaptive…