Flatness vs LRS

[mathkook]

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.

[mathkook]

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.