Encoding superpermutations

[Kyle Gullion]

I've been messing around with various transformations of superpermutations, and what I'm calling the move-to-back transformation is one that really jumps out as potentially useful. I go over why I think that in this notebook (scroll past all the code if you're not into that kind of thing) but in a nutshell, move-to-back is the move-to-front transform in reverse and turns this:

012340213402314023410243102413024103241023401230412301423012430120341203142031240312043120134201324013204132014321042310421304210342104321403214302143201

into this:

000000100000100000100010000010000010000010010001000001000001000002100000100000100000100000210000010000010000010001001000001000001000001000100000100000100

and the encoder state transitions look like this:

I've also hacked together a small app to mess around with traversing the encoder states here: https://kgullion.github.io/superm/

The app should be fairly self explanatory after looking over the notebook but be sure to scroll to the bottom where you'll find the physics model settings.

I've had a lot of fun messing around with this and hopefully it can contribute to some new ideas!

[Miles Gould]

Very cool! Is it possible to use the layout from one superpermutation when rendering another, so you can see the differences in how they traverse the 1-cycles? I'm intrigued by the line "we end up with something that looks more like someone chopped up one side of SP4 before replacing the vertices then just glued everything back together" - it might be easier to visualise that sort of thing with a consistent layout.

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[Kyle Gullion]

Disabling the physics engine after doing the first layout will allow a new encoding to be copy and pasted in without moving any nodes. To better illustrate the pattern I'm seeing, I've highlighted the SP4 "core" in SP5,0 below and left everything except the traces for SP5,2. I've also completed the pattern with a '21' or a '321' on the end of the encodings.

SP5,0 (encoding used 000000000100000100000100000210000010000010000010000021000001000001000001000003210000010000010000010000021000001000001000001000002100000100000100000100000321):

SP5,2 (encoding used 00000000010000010000010000021000001000001000001000100100000100000100000100010000010000010021001000001000001000100000100000100000100100010000010000010000021):

In both encodings, the left side is completely identical, while a different set of cycles are used on the right. Each rank1 cycle (blue lines) has matching node colors to make it a bit easier to see where each line is actually going in SP5,2. The edges that are fixed in place end up being traversed on different indices between each encoding and yet the same set of fixed edges appears in every SP5.