Probable feature of an n = 8 superpermutation beating the upper bound

[Randy Olsen]

In Robin Houston's n = 6 superpermutation of length 872 I've been studying (the particular solution currently on the OEIS page), three permutations are utilized differently than all the others (they start with 651, 652, and 653, but the particular permutations are not of primary interest to me at the moment.)

Similarly, in Greg Egan's n = 7 superpermutation of length 5906 he posted to his page, seventeen permutations are utilized differently than the others (and they all seem to start with 73, 75, or 76.)

For an n = 8 superpermutation that would have length under the upper bound of 46205, I would like to make a case for expecting around 100 of these specially utilized permutations, and specifically my favored contender is 106, as I will explain, but I would invite others to present counterarguments for this number being too high or too low.  I said "around 100" because another contender is 95, but I'm open to having my misunderstanding corrected if even "around 100" would be way off for some reason. 

The three from n = 6 and the seventeen from n = 7 are the permutations that, in my "negative space" terminology, are the ones each associated with contributing a chunk of negative space of value 2*n toward the total.  

Am I equivalently just finding the permutations associated with what most call weight-3 edges?

Regardless, except for the end rectangle contributing n*(n-1) to the negative space total, all of the other negative space in these n = 6 and n = 7 superpermutations arrives in chunks of just n.  Unlike the n = 5 case, which has one chunk of 3*n, these n = 6 and n = 7 superpermutations get by without any chunks 3*n.

Assuming an n = 8 superpermutation that would beat the upper bound can get by similarly, the question becomes the division of negative space chunks of just n and chunks of 2n.

As usual, I have done my spreadsheet explorations, plus a semi-systematic check of OEIS for corresponding sequences, and I would characterize a respective n = 6, n = 7, and n = 8 sequence of 3, 17, 106, or 3, 17, 95 for the number of appearances of the "chunk 2*n" permutations as well-behaved and suggesting continued well-behaved results through at least n = 10.  

Why 106 is my favored contender is because searching for 3, 17, 106 in OEIS reveals a fascinating sequence about rooted trees, with an example diagram not so unlike the tree diagrams on Greg Egan's page about superpermutations:  

https://oeis.org/A299039

However, I will allow that this is only a candidate, and that 95 is also a contender because the sequence would then be defined by (n-4)^2*([n-4]-1)! - 1.

Here is my spreadsheet with these explorations:

https://1drv.ms/x/s!Ap4I-rXcEKcQjXYsqwwHtXVOKr9p?e=2yfqP6

In addition, here is my spreadsheet work for Robin Houston's n = 6 superpermutation "negative space" calculation:

https://1drv.ms/x/s!Ap4I-rXcEKcQjXJPZhOHy4ZZumHW?e=m5DHUL

Here is my spreadsheet work for Greg Egan's n = 7 superpermutation "negative space" calculation:

https://1drv.ms/x/s!Ap4I-rXcEKcQjXTxTkPHCmPJednX?e=fzoUwO

Lastly, based on my previous posting I would caution that a secondary correction could kick in at n = 10, as in my "negative space" chain I believe I have equivalently established that *if* the minimal superpermutation length [or a new lower bound] goes as "n-5 below upper bound" for n = 6 and subsequent, meaning 872, 5906, 46202, 408962, 4037042, etc, then a connection to OEIS sequence A288780 exists through a correction that gains a predictable new term each time n is incremented by 4: minlen(n-4) shows up in the first round, then minlen(n-4) and minlen(n-8) in the second round, then minlen(n-4), minlen(n-8), and minlen(n-12) in the third round, and so forth.  At least, you could call that a conjecture, as I have not gone and revised that previous post's spreadsheet with explicit calculations to establish that better.