This is an answer for question posted in one of the comments.
Sorry, I don't know what a "cubically suspended" strict omega-category is, so I do not know anything related. I don't know of any special orderings on the set you describe, but you can create tons and tons of artificial ones.
One way is to look for distances and norms for permutations and then design function $f$ that combines them into $g(x) = f(d(x,112233),d(x,123123))$ such that your condition holds. Trivial example is $f(d_1, d_2) = \langle {d_1 \leq 0, d_2 > 0} \rangle$ with standard partial order on $\{true, false\}^2$.
You may also think about those sequences as two copies of set $\{1, \ldots, n\}$ painted, say, red and green (in a sequence ...4...4...
, the first symbol 4
is red and the second is green). Then, if you consider the set of indices of green numbers, your minimal and maximal elements are the boundaries, you cannot get smaller than in 112233...
, you can get greater than in 123..123..
. Just add some set ordering, I think there are many that will fit.
One more way is that you could split your sequence in half and construct step-differences, i.e. 010 010
for 1122 3344
and 111 111
for 1234 1234
, and then apply some even simple metrics.
I know I haven't answered your question--those are just some examples. However, I hope it will be useful to you, even if only a bit.