Since flatness and longest-repeated-substring (LRS) were both mentioned in posts on these boards, I thought I’d see how (or if) the two are related related.
Here’s a chart for all 99 rational cycles of length k=13 with x=8 odds. Flatness is the ratio of the largest odd member to the smallest odd member, and LRS is the longest repeated substring in the circular parity sequence (up-down-up-up-down-…). That’s the high cycle in the upper left and the circuit in the middle right.
Here’s a similarly-shaped graph for k=18, x=11. (Flatter cycles are on the left, not right.)
And here’s an annotated one for the 9,690 rational cycles with k=21, x=13.
I wonder if there is a way to characterize where a dots are allowed to appear on scatterplots like this (in terms of k and x). Some areas seem verboten.
A more specific question, about de Bruijn-like cycles (lower left) with minimal LRS … are there upper and lower bounds on how flat they can be (in terms of k and x)?
PS. Recap of this chart: The high cycle (upper left) has the minimal flatness and the maximal LRS length (k-2), and it is also a k-x cycle (no adjacent even terms). The circuit (middle right) has maximal flatness and larger than average LRS (x-1). The de Bruijn-like cycles have minimal LRS (O(\log k)).
