Riffs & Rotes & A061396

[Jon Awbrey]

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SeqFan Electorate,

Here are a couple more pages from my notes,
where it looks like I first arrived at the
generating function, and also carried out
some brute force enumerations of riffs.

I am going to experiment with a different way of
transcribing indices and powers into a plaintext.

|                jj
|              p<
|      j      /  ji
|    p<     p<         etc.
|      i      \  ij
|              p<
|                ii

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1978-11-06

Generating Function

| R(x) = 1 + x + 2x^2 + ...
|
|      =   1 + x.x^0 (1 + x + 2x^2 + ...)
|        . 1 + x.x^1 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . 1 + x.x^2 (1 + x + 2x^2 + ...)
|        . ...
|
|      = 1 + x + 2x^2 + ...
|
| Product over (i = 0 to infinity) of (1 + x.x^i.R(x))^R_i  =  R(x)

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1978-11-10

Brute force enumeration of R_n

| 4 p's
|
|       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
|     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<        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_p   p p<
|               p^p
|

Altogether, 20 riffs of weight 4.

| o---------------------o---------------------o---------------------o
| | 3                   | 4                   | 5                   |
| o---------------------o---------------------o---------------------|
| | // // 2             | 10, 3, 1, 6         | 36, 10, 2, 3, 2, 20 |
| o---------------------o---------------------o---------------------|
| |                     | 0^1 4^1,            |                     |
| |                     | 1^1 3^1,            |                     |
| |                     | 2^2,                |                     |
| |                     | 4^1 0^1             |                     |
| o---------------------o---------------------o---------------------o
| | 6                   | 20                  | 73                  |
| o---------------------o---------------------o---------------------o
|
| [Sloane's Handbook 1973]
|
| #644 ~~ 1, 2, 6, 20, 71, 259, 961
|         perms by inversions
|         Netto: Lehrbuch Combk 2nd ed. 1927 p.96
|
| #645 ~~ 1, 2, 6, 20, 76, 312, 1384
|         symmetric permutations
|         Lucas: Théo Nombres v.1 p.221 (1891)

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Here are the number values of the riffs on 4 nodes:

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

For ease of reference, I include the previous table
of smaller riffs and rotes, redone in the new style.

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