Riffs & Rotes & A061396

[Jon Awbrey]

¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤

Neil,

I wanted to try and do this trick for riffs
that you did with Goebel's correspondence:

Triangle in which k-th row lists natural number
values for the collection of riffs with k nodes.

k | natural numbers n such that |riff(n)| = k
--o------------------------------------------------
0 | 1;
1 | 2;
2 | 3, 4;
3 | 5, 6, 7, 8, 9, 16;
4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27,
  | 32, 49, 53, 64, 81, 128, 256, 512, 65536;

The natural number values for the riffs with
at most 3 pts are as follows (@'s are roots):

|                 o       o  o       o
|                 |       ^  |       ^
|                 v       |  v       |
|           o  o  o    o  o  o  o o  o
|           |  ^  |    |  |  ^  | ^  ^
|           v  |  v    v  v  |  v/   |
| Riff:  @; @, @; @, @ @, @, @, @,   @;
|
| Value:  2; 3, 4; 5,  6 , 7, 8, 9, 16;

---------------------------------------------------

Is this the right idea?

Jon

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Tables for Reference:

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<        @     o      @            ((((())))())
|                               p^p      ^   ^
|                                         \ /
|                                          o
|
|                                                        o---o
|                                                        |
|                                                        o---o
|                                          o             |
| 8         p_1^3 =                       ^ ^        o---o
|           p_1^p_2 =           p_p      /   \       |
|           p_1^p_(p_1)       p<        @     o      @            ((((())())))
|
|                                                    o-o o-o
|                                          o         |   |
| 9         p_2^2 =                       ^          o---o
|           p_(p_1)^2 =         p        /           |
|           p_(p_1)^(p_1)     p<        @            @            ((())((())))
|                               p        ^
|                                         \
|                                          o
|
|                                             o              o---o
|                                            ^               |
|                                           /            o---o
|                                          o             |
| 16        p_1^4 =               p       ^          o---o
|           p_1^(p_1^2) =       p<       /           |
|           p_1^(p_1^p_1)     p<        @            @            (((((())))))
|
o--------------------------------------------------------------------------------

o--------------------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
|   p<        p<        p p_p     p<_p     p_p_p     p_p<
| p<        p<        p<        p<       p<        p<
|
| 2^16      2^8       2^6       2^9      2^5       2^7
| 65536     256       64        512      32        128
o--------------------------------------------------------------------------------
|
|       p
|     p<        p_p                 p                    p
| p_p<      p_p<      p<        p_p<_p   p_p_p_p   p_p_p<
|                       p p_p
|
| p_16      p_8       p_6       p_9      p_5       p_7
| 53        19        13        23       11        17
o--------------------------------------------------------------------------------
|
|     p
|   p<        p_p       p         p                    p
| p<        p<        p<        p<       p^p p_p   p p<
|   p         p         p_p       p^p                  p
|
| 3^4       3^3       5^2       7^2
| 81        27        25        49       12        18
o--------------------------------------------------------------------------------
|
| p p_p_p   p p<
|               p^p
|
| 10        14 
o--------------------------------------------------------------------------------