Riffs & Rotes & A061396

[Jon Awbrey]

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SeqFan Danglers,

This business of putting things in triangles,
that Neil applied to the Göbel numberings of
rooted trees, has reminded me of some of the
reasons that I strung myself out between the
riffs and the rotes for all those many years,
to wit, it was in asking myself the question:
How much of this order of natural numbers is
purely combinatorial?  By that I meant:  How
much of the familiar, all too familiar order
of the sequence 1, 2, 3, ... can be regarded
as based totally on combinatorial parameters
and properties of some associated graph-like
structures?

For instance, one can say that 1 < 2 because
rote(1) is strictly engraphed within rote(2),
that is to say, is naturally embedded as its
proper subgraph.  But then we are obliged to
ask ourselves why should we prefer the order
3 < 4 over that of 4 < 3, and so on down the
line of succession of all the other naturals.
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 multipicative structure?

Failing to derive the entire linear order of
the natural numbers, as one must most likely,
upon such paltry strictures, one may yet ask:
What ilk of order structure is so determined?
How much is combinatorial;  How much residue?
But that is too many questions for a weekend.

That was the general idea, anyway, even if I
have most likely forgotten one or two of the
details that I would need in order to render
this query a sensible congeries of questions.

Jon Awbrey

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o--------------------------------------------------------------------------------
| integer   factorization     riff      r.i.f.f.     rote   -->   in parentheses
|                             k p's     k nodes      2k+1 nodes
o--------------------------------------------------------------------------------
|
| 1         1                 blank     blank        @            blank
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
| 2         p_1^1             p         @            @            (())
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
| 3         p_2^1 =                                  |
|           p_(p_1)^1         p_p       @            @            ((())())
|                                        ^
|                                         \
|                                          o
|
|                                                        o---o
|                                          o             |
|                                         ^          o---o
| 4         p_1^2 =                      /           |
|           p_1^p_1           p^p       @            @            (((())))
|
o--------------------------------------------------------------------------------
|
|                                                    o---o
|                                                    |
|                                                    o---o
|                                                    |
| 5         p_3 =                                    o---o
|           p_(p_2) =                                |
|           p_(p_(p_1))       p_(p_p)   @            @            (((())())())
|                                        ^
|                                         \
|                                          o
|                                           ^
|                                            \
|                                             o
|
|                                                        o-o
|                                                       /
|                                                  o-o o-o
| 6         p_1 p_2 =                               \ /
|           p_1 p_(p_1)       p p_p     @ @          @            (())((())())
|                                          ^
|                                           \
|                                            o
|
|                                                        o---o
|                                                        |
|                                                    o---o
|                                                    |
| 7         p_4 =                                    o---o
|           p_(p_1^2) =                              |
|           p_(p_1^p_1)       p_(p^p)   @     o      @            ((((())))())
|                                        ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =                    /   \       |
|           p_1^p_(p_1)       p^p_p     @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =                  /           |
|           p_(p_1)^(p_1)     p_p^p     @            @            ((())((())))
|                                        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =                       ^          o---o
|           p_1^(p_1^2) =                /           |
|           p_1^(p_1^p_1)     p^(p^p)   @            @            (((((())))))
|
o--------------------------------------------------------------------------------