Pairwise order of a sequence of elements

In the very first post of this blog, I described a new measure of disorder which I called \(\mathit{Amp}\). I started its construction process by introducing what I dubbed the pairwise order of a sequence as follows: A few days ago I went back to one of the simplest possible tools in the domain: a three-way comparator for two values: \[\mathit{comp}(x, y)= \begin{cases} 1 & \text{ if } x \lt y\\ -1 & \text{ if } x \gt y\\ 0 & \text{otherwise} \end{cases}\] Nothing ground-breaking so far, it’s similar to a negation of the sign function applied to the difference of two real numbers. Let’s apply it to all adjacent pairs of elements of a sequence of elements \(X\): Still fairly boring if I’m being honest. We will call that new sequence the pairwise order of the sequence for the rest of the article. Today’s article is all about showing that this so-called pairwise order is, in fact, maybe not all that boring. Definition(s) and properties The definition above was spot-on when talking about…