Question 1 is clearly the usual stars-and-bars question (with solution ). As for question 3, a quick back-of-the-envelope estimate – assuming that a generic string differs from all its shifts and reflections (which is true for something like strings) – gives that the amount of strings corresponding to a given orthobrace is linear in the length of the orthobrace, thus making the amount of orthobraces grow about as . This, in turn, suggests that the problem you’re alluding to in question 4 is that we we need to canonicize an exponentially-growing set by a linearly-growing relation, which would take exponential time. Hence the need for a canonical-by-construction generating function.

One thing that does stand out to me is that the identifications you made (orthobraces are considered identical up to cyclic permutations and reversals (which corresponds to mirroring the orthogon)) correspond to quotienting the set of strings over by the action of the infinite dihedral group.

In more detail, let the set of all strings of length over , and .

Then acts on by sending to a one-symbol shift (in an arbitrary direction) and to reversal.

To extend these actions to an action on , we first view as , which clearly extends to an action by for all by the natural map . This procedure extends to , giving an action of on for all .

Taking the union of these actions, we obtain an action on .

Quotienting by this action, we obtain the braces,

with the orthobraces being just those braces arising from under the action.

This gives a way to improve the estimate of the growth rate from above, by use of Burnside’s theorem. Clearly it suffices to estimate the growth rate at prime lengths (used to keep the strings fixed by manageable$).

Noting that fixes concatenations of palindromes of lengths – of which there are and that fixes strings that are repetitions of strings of length for , we have that the amount of braces – which equals the average amount of points fixed by an action of – is – the same estimate, but with better constants.

P.S. Sorry for the more rambly nature of this comment, I didn’t take the time to properly digest my thoughts here.

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