Riffs & Rotes & A061396

[Marc LeBrun]

 > Jon Awbrey
 > This whole genus of questions is what I mean
 > by:  Is there any "purely combinatorial" (PC)
 > reason for the usual ordering of the natural
 > numbers, one that bases itself wholly on the
 > indicated indices of multiplicative structure?

I certainly haven't been able to absorb as much of this interesting 
material as I'd like, but if I've followed this much correctly I think the 
answer to this question must be "no".

The multiplicative structures are independent of the actual values of the 
primes.  If you were to swap 2 and 3, say, certain trees would just map 
onto different integers.  But the set of all trees would still cover all 
the integers.

Moreover, the primes involved might just as well be those of, say, the 
Gaussian integers or other such extension.  Then, of course, the idea 
of  ordering itself becomes tricky.