Riffs & Rotes & A061396

[Jon Awbrey]

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SeqFanTasians,

I would like to step back a bit from my retrospective
and try to make some connections with my present work,
specifically for the reason that I think that this is
where some new enumerative problems might be found to
arise, all of which I no longer have the proper tools
to tackle, but I think that some of this community is
almost bound to.  During the 80's my work shifted its
focus from the enumerative to the computational.  Yes,
there was a distinction then, strange to say, and yet
what I meant was more toward theorem-proving programs.
All along I had been keeping an eye out for relations
among arithmetic (number theory) and algebra (groups),
geometry (graphs), and logic -- the "lambda point" as
one of my professors (Davis) called it then, counting
the first two as one.  So the little bit of graph and
group theory that I picked up was already ready to be
converted into data-structures and algorithms able to
take advantage of their symmetries, all in support of
the new logical mission.  But I had to let go of what
I earnestly would have kept, all the joys of counting.

 From my present perspective, what all of these graphs
and other sorts of structures have in common is their
so-called "reflective" or "self-documenting" property.
The archetype of these "reflective or auto-documental"
data structures is, of course, the usual construction
of natural numbers as sets -- {}, {{}}, {{}{{}}}, ... --
which builds its own distinctive character by keeping
a record of its history as it goes.  In a similar way,
I recall that I first began to think of riffs and all
their kin by asking myself a question about molecular
tagging:  What would be the limit of a tagging tactic?
For now, I do not think that I can explain myself any
better, but I hope the sense of it will clear in time.

Enough flash-forward -- back to the story ...

The earliest indication of this line of thinking,
that I can find at present among my personal records,
is contained in the following letter to Martin Gardner:

| 1976-08-16
|
| Mr. Martin Gardner
| Scientific American
|
| Dear Sir:
|
| On the following sheet find the beginnings of a correspondence
| between integers and petrified trees, those so rooted and embedded
| in their strata as to have a fixed order of limbs at each node.
| Read blank areas as nodes, the outside being counted first, and
| enclosure relations as branches.  The 0 is merely an orthographic
| variant for a final parenthesis, reading inward.
|
| ...
|
| This arrangement of trees I have dubbed "the forest primeval".
| It is developed by expressing each integer in its prime factorization,
| and then carrying this process to its extremes, factoring each exponent
| into powers of primes, and each of these powers in turn, quitting only
| when zero (or empty sets of) exponents are reached, and given the convention
| that unit exponents be expressed as 2^0.  This suggests a system of numerals
| with primes in order as place values, the place-holders being exponents instead
| of multipliers, the held values being multiplied instead of added together,
| and with the same multitude of place-values implied in each place.
|
|                                                                             o
|                                                                            /
|                                                                       o   o   o
|                                                                        \ /   /
|                                                                         o   o
|                                                                          \ /
| Example:  24  =  2^3 3^1  =  2^(2^0 3^(2^0)) 3^(2^0)  =  (()(()))(())  =  @
|
| I have therefore sometimes thought of this as a notation in which numerals are
| wholly expressed by means of place-value, there being but one symbol, whose only
| office is to reserve a place;  but I'm not sure how much this means, since as much
| might be said of simple tally marks, with alot less complexity taken for granted.
|
| [Enclosure:]
|
|  1     0
|  2     (0)
|  3     0(0)
|  4     ((0))
|  5     00(0)
|  6     (0)(0)
|  7     000(0)
|  8     (0(0))
|  9     0((0))
| 10     (0)0(0)
| 11     0000(0)
| 12     ((0))(0)
| 13     00000(0)
| 14     (0)00(0)
| 15     0(0)(0)
| 16     (((0)))
|
| ...    ...

Continuing apologies for dragging you through my memoirs
this way, but there are just too many important details
and incidental sidelights that I will otherwise forget.

Jon Awbrey

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