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pjs36
pjs36
00:30
So I should have found the orbits ten times over by now, but GAP keeps crashing (and Sage does almost all of its group theory stuff by borrowing GAP stuff, from what I can tell). More accurately, I apparently overstay my welcome. I get error messages like "the session in which this object was defined is no longer running." I'm sure I'll be able to find a way around it, but it's crashed in the middle of the 3rd orbit several times now...
 
13 hours later…
arctic tern
arctic tern
13:19
huh, I guess it's a well-known graph theory problem and hasn't been solved even for n=6
pjs36
pjs36
13:47
Yeah, I felt like I should have known enough about it to know that it was the same problem you were asking about, but I wasn't really familiar with it.
Did you end up finding anything about orbits? Still fighting the computer over here, to an almost comical degree...
 
2 hours later…
pjs36
pjs36
15:27
Alas, it was amateur programming, yet again: I was doing something for each total in a set of remaining totals (those whose orbit had not yet been found), while making the set of remaining totals smaller with each iteration. It would seem the set of remaining totals in the initial for loop was held constant, so that even totals whose orbit had been found, were being investigated again. An easy fix, once I suspected it was the problem!
pjs36
pjs36
15:38
For n = 4, it would seem the orbit sizes are 2520, 1680, 960, 630, 420, 30, with corresponding stabilizer sizes of 16, 24, 42, 64, 96, 1344. I have representatives from each orbit too. Thankfully I printed them out as the computation ran, because it would seem my method of saving things is acting funny...
pjs36
pjs36
16:07
And of course, the number of orbits was referenced in the OEIS page on the number of totals...
pjs36
pjs36
16:51
Orbit of size 2520
Representative total (which must have stabilizer of size 16)
{{8, 6}, {2, 4}, {1, 5}, {3, 7}}
{{5, 6}, {2, 3}, {4, 7}, {8, 1}}
{{1, 2}, {3, 6}, {8, 7}, {4, 5}}
{{6, 7}, {2, 5}, {8, 3}, {1, 4}}
{{1, 7}, {2, 6}, {3, 5}, {8, 4}}
{{5, 7}, {3, 4}, {8, 2}, {1, 6}}
{{1, 3}, {8, 5}, {4, 6}, {2, 7}}
Orbit of size 1680
Representative total (which must have stabilizer of size 24)
{{8, 7}, {1, 5}, {2, 3}, {4, 6}}
{{5, 6}, {8, 1}, {3, 4}, {2, 7}}
{{8, 2}, {3, 5}, {1, 6}, {4, 7}}
{{1, 2}, {6, 7}, {8, 3}, {4, 5}}
{{2, 4}, {8, 5}, {1, 7}, {3, 6}}
{{8, 6}, {3, 7}, {1, 4}, {2, 5}}
{{1, 3}, {5, 7}, {2, 6}, {8, 4}}
Orbit of size 960
Representative total (which must have stabilizer of size 42)
{{1, 7}, {2, 6}, {3, 5}, {8, 4}}
{{2, 3}, {8, 5}, {6, 7}, {1, 4}}
{{8, 7}, {1, 6}, {2, 5}, {3, 4}}
{{2, 4}, {5, 6}, {3, 7}, {8, 1}}
{{1, 5}, {8, 2}, {4, 7}, {3, 6}}
{{5, 7}, {1, 2}, {4, 6}, {8, 3}}
{{8, 6}, {1, 3}, {2, 7}, {4, 5}}
Orbit of size 630
Representative total (which must have stabilizer of size 64)
{{3, 5}, {2, 6}, {1, 7}, {8, 4}}
{{6, 7}, {1, 2}, {8, 5}, {3, 4}}
{{5, 7}, {2, 4}, {8, 3}, {1, 6}}
{{8, 2}, {1, 5}, {4, 6}, {3, 7}}
{{5, 6}, {2, 3}, {1, 4}, {8, 7}}
{{1, 3}, {8, 6}, {4, 5}, {2, 7}}
{{4, 7}, {3, 6}, {2, 5}, {8, 1}}
Orbit of size 420
Representative total (which must have stabilizer of size 96)
{{1, 5}, {8, 2}, {4, 7}, {3, 6}}
{{8, 7}, {2, 6}, {1, 4}, {3, 5}}
{{3, 4}, {6, 7}, {2, 5}, {8, 1}}
{{1, 3}, {5, 6}, {8, 4}, {2, 7}}
{{2, 4}, {3, 7}, {8, 5}, {1, 6}}
{{8, 6}, {2, 3}, {1, 7}, {4, 5}}
{{5, 7}, {1, 2}, {4, 6}, {8, 3}}
Orbit of size 30
Representative total (which must have stabilizer of size 1344)
{{1, 2}, {3, 5}, {4, 6}, {8, 7}}
{{8, 2}, {3, 6}, {1, 7}, {4, 5}}
{{2, 4}, {3, 7}, {8, 5}, {1, 6}}
{{8, 6}, {1, 5}, {2, 3}, {4, 7}}
{{1, 3}, {8, 4}, {6, 7}, {2, 5}}
{{5, 7}, {2, 6}, {1, 4}, {8, 3}}
{{5, 6}, {2, 7}, {8, 1}, {3, 4}}
(Sorry for bombarding the chat, but I thought it might be of relative interest, and didn't really have a better way)
pjs36
pjs36
17:48
And just for fun, a colored total i.sstatic.net/sDexQ.png
 
3 hours later…
pjs36
pjs36
20:40
And I think I realized why none of the totals have full dihedral symmetry. Any permutation in Sym(8) that stabilizes a total necessarily permutes the seven synthemes. But $S_7$ has no elements of order 8. (I only realized this after seeing checking each of the 6240 totals, to see if any were stabilized by an 8-cycle...)
There are some with symmetries-of-the-square dihedral symmetry, though.

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