User:Jon Awbrey/WORKSPACE

Articles

Logic Syllabus

Precursors Of Category Theory

Graph Theory, Arithmetic
- Forest Primeval
- Riffs and Rotes

Graph Theory, Computation
- Propositions As Types Analogy
- Differential Analytic Turing Automata

Graph Theory, Logic

Differential Logic

Functional Logic

Information Theory

User Pages

Syllabus

Logical operators

Logical concepts

Relational concepts

Work Notes

I thought I might start by upgrading the graphics for some of my sequences from ASCII to EPS, GIF, JPG, PNG, whatever works best here — maybe SVG if I can find an easy enough graphics package that works for that. Jon Awbrey 02:30, 31 October 2009 (UTC)

Other Workspaces

User:Jon Awbrey/Figures and Tables

1 Articles
2 User Pages
3 Syllabus
- 3.1 Logical operators
- 3.2 Logical concepts
- 3.3 Relational concepts
- 3.4 Related articles
4 Work Notes
5 A061396
- 5.1 Wiki + TeX + JPEG
- 5.2 ASCII
6 A062504
- 6.1 TeX Array
- 6.2 JPEG
- 6.3 ASCII
7 A062537
- 7.1 Wiki + TeX + JPEG
8 A062860
- 8.1 Wiki + TeX + JPEG
9 A106177
- 9.1 Primal Codes of Finite Partial Functions on Positive Integers
- 9.2 Wiki Table
- 9.3 Wiki + TeX
- 9.4 ASCII
10 A106178
- 10.1 Wiki Table
- 10.2 TeX Smallmatrix
- 10.3 ASCII
11 A108352
- 11.1 Links
- 11.2 TeX Array
- 11.3 ASCII
12 A108353
- 12.1 TeX Array
- 12.2 ASCII
13 A108370
14 A108371
- 14.1 Wiki Table
- 14.2 ASCII
15 A108372
16 A108373
17 A108374
- 17.1 TeX Array
- 17.2 ASCII
18 A109297
- 18.1 TeX Array
- 18.2 ASCII
19 A109298
- 19.1 TeX Array
- 19.2 ASCII
20 A109299
- 20.1 TeX Array
- 20.2 ASCII
21 A109300
- 21.1 JPEG
- 21.2 ASCII
22 A109301
- 22.1 Example
- 22.2 JPEG
- 22.3 ASCII
23 A111788
24 A111789
25 A111790
26 A111791
- 26.1 TeX Array
- 26.2 Wiki Table
- 26.3 ASCII
27 A111792
- 27.1 TeX Array
- 27.2 ASCII
28 A111793
- 28.1 TeX Array
- 28.2 ASCII
29 A111794
- 29.1 TeX Array
- 29.2 Wiki Table
- 29.3 ASCII
30 A111795
- 30.1 JPEG
- 30.2 ASCII
31 A111796
- 31.1 TeX Array
- 31.2 ASCIII
32 A111797
- 32.1 TeX Array
- 32.2 ASCII
33 A111798
- 33.1 TeX Array
- 33.2 ASCII
34 A111799
- 34.1 TeX Array
- 34.2 ASCII
35 A111800
- 35.1 TeX + JPEG
- 35.2 ASCII
36 A111801
37 A112095
- 37.1 TeX Array
- 37.2 ASCII
38 A112096
- 38.1 TeX Array
- 38.2 ASCII
39 A112480
- 39.1 TeX Array
- 39.2 ASCII
40 A112481
- 40.1 TeX Array
- 40.2 ASCII
41 A112846
42 A112868
- 42.1 TeX Array
- 42.2 ASCII
43 A112869
- 43.1 TeX Array
- 43.2 ASCII
44 A112870
- 44.1 TeX Array
- 44.2 ASCII
45 A112871
- 45.1 TeX Array
- 45.2 ASCII
46 A112872
47 A113197
- 47.1 TeX Array
- 47.2 ASCII
48 A113198
- 48.1 TeX Array
- 48.2 ASCII
49 A113199
- 49.1 TeX Array
- 49.2 ASCII
50 A113200
- 50.1 TeX Array
- 50.2 ASCII

A061396

Number of "rooted index-functional forests" (Riffs) on n nodes. Number of "rooted odd trees with only exponent symmetries" (Rotes) on 2n+1 nodes.

Wiki + TeX + JPEG

{\text{Prime Factorizations, Riffs, Rotes, and Traversals}}\!

{\text{Integer}}\!

{\text{Factorization}}\!

{\text{Notation}}\!

{\text{Riff Digraph}}\!

{\text{Rote Graph}}\!

{\text{Traversal}}\!

1\!

1\!

2\!

{\text{p}}_{1}^{1}\!

{\text{p}}\!

((~))

$3\!$	${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}_{\text{p}}\!$			$(((~))(~))$
$4\!$	${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}^{\text{p}}\!$			$((((~))))$

$5\!$	${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$	$((((~))(~))(~))$
$6\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{\text{p}}\!$	$((~))(((~))(~))$
$7\!$	${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[10pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$	$(((((~))))(~))$
$8\!$	${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$	$(((((~))(~))))$
$9\!$	${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{\text{p}}^{\text{p}}\!$	$(((~))(((~))))$
$16\!$	${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[10pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}^{\text{p}}}\!$	$((((((~))))))$

ASCII

Illustration of initial terms of A061396
Jon Awbrey (jawbrey(AT)oakland.edu)

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

Further Comments:

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

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

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)

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

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
|

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

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

(later)

Expanded version of first table:

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      2^9        2^8       2^7       2^6       2^5
| 65536     512        256       128       64        32
|
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_9        p_8       p_7       p_6       p_5
| 53        23         19        17        13        11
|
o--------------------------------------------------------------------------------
|
|   p^p       p_p        p         p
| p<        p<         p<        p<
|   p         p          p^p       p_p
|
| 3^4       3^3        7^2       5^2
| 81        27         49        25
|
o--------------------------------------------------------------------------------
|
|     p
| p p<      p p<       p^p p_p   p p_p_p
|     p         p^p
|
| 18        14         12        10
|
o================================================================================

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;

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

1, 2, 3, 4, 5, 6, 7, 8, 9, 16,
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536,

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

1; 2; 3, 4; 5, 6, 7, 8, 9, 16;
10, 11, 12, 13, 14, 17, 18, 19,
23, 25, 27, 32, 49, 53, 64, 81,
128, 256, 512, 65536;

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

A062504

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

TeX Array

${\begin{array}{l|l|r}k&P_{k}=\{n:\operatorname {riff} (n)~{\text{has}}~k~{\text{nodes}}\}=\{n:\operatorname {rote} (n)~{\text{has}}~2k+1~{\text{nodes}}\}&|P_{k}|\\\hline 0&\{1\}&1\\1&\{2\}&1\\2&\{3,4\}&2\\3&\{5,6,7,8,9,16\}&6\\4&\{10,11,12,13,14,17,18,19,23,25,27,32,49,53,64,81,128,256,512,65536\}&20\end{array}}$

JPEG

{\text{Prime Factorizations, Riffs, and Rotes}}\!

{\text{Integer}}\!

{\text{Factorization}}\!

{\text{Notation}}\!

{\text{Riff Digraph}}\!

{\text{Rote Graph}}\!

1\!

1\!

2\!

{\text{p}}_{1}^{1}\!

{\text{p}}\!

$3\!$	${\begin{array}{lll}{\text{p}}_{2}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}_{\text{p}}\!$
$4\!$	${\begin{array}{lll}{\text{p}}_{1}^{2}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}^{\text{p}}\!$

$5\!$	${\begin{array}{lll}{\text{p}}_{3}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$
$6\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{\text{p}}\!$
$7\!$	${\begin{array}{lll}{\text{p}}_{4}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$
$8\!$	${\begin{array}{lll}{\text{p}}_{1}^{3}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$
$9\!$	${\begin{array}{lll}{\text{p}}_{2}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{\text{p}}^{\text{p}}\!$
$16\!$	${\begin{array}{lll}{\text{p}}_{1}^{4}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{2}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}^{\text{p}}}\!$

$10\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{3}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$
$11\!$	${\begin{array}{lll}{\text{p}}_{5}^{1}&=&{\text{p}}_{{\text{p}}_{3}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{2}^{1}}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$
$12\!$	${\begin{array}{lll}{\text{p}}_{1}^{2}{\text{p}}_{2}^{1}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}\end{array}}$	${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}\!$
$13\!$	${\begin{array}{lll}{\text{p}}_{6}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$
$14\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{4}^{1}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{2}}^{1}\\[12pt]&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$
$17\!$	${\begin{array}{lll}{\text{p}}_{7}^{1}&=&{\text{p}}_{{\text{p}}_{4}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{2}}^{1}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$
$18\!$	${\begin{array}{lll}{\text{p}}_{1}^{1}{\text{p}}_{2}^{2}&=&{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!$
$19\!$	${\begin{array}{lll}{\text{p}}_{8}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{3}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{2}^{1}}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$
$23\!$	${\begin{array}{lll}{\text{p}}_{9}^{1}&=&{\text{p}}_{{\text{p}}_{2}^{2}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!$
$25\!$	${\begin{array}{lll}{\text{p}}_{3}^{2}&=&{\text{p}}_{{\text{p}}_{2}^{1}}^{{\text{p}}_{1}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!$
$27\!$	${\begin{array}{lll}{\text{p}}_{2}^{3}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!$
$32\!$	${\begin{array}{lll}{\text{p}}_{1}^{5}&=&{\text{p}}_{1}^{{\text{p}}_{3}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{2}^{1}}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$
$49\!$	${\begin{array}{lll}{\text{p}}_{4}^{2}&=&{\text{p}}_{{\text{p}}_{1}^{2}}^{{\text{p}}_{1}^{1}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{{\text{p}}_{1}^{1}}\end{array}}$	${\text{p}}_{{\text{p}}^{\text{p}}}^{\text{p}}\!$
$53\!$	${\begin{array}{lll}{\text{p}}_{16}^{1}&=&{\text{p}}_{{\text{p}}_{1}^{4}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{2}}}^{1}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}}^{1}\end{array}}$	${\text{p}}_{{\text{p}}^{{\text{p}}^{\text{p}}}}\!$
$64\!$	${\begin{array}{lll}{\text{p}}_{1}^{6}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}{\text{p}}_{2}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{1}{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}{\text{p}}_{\text{p}}}\!$
$81\!$	${\begin{array}{lll}{\text{p}}_{2}^{4}&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{2}}\\[12pt]&=&{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}_{\text{p}}^{{\text{p}}^{\text{p}}}\!$
$128\!$	${\begin{array}{lll}{\text{p}}_{1}^{7}&=&{\text{p}}_{1}^{{\text{p}}_{4}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{2}}^{1}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}^{1}}\end{array}}$	${\text{p}}^{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$
$256\!$	${\begin{array}{lll}{\text{p}}_{1}^{8}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{3}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{2}^{1}}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$
$512\!$	${\begin{array}{lll}{\text{p}}_{1}^{9}&=&{\text{p}}_{1}^{{\text{p}}_{2}^{2}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{{\text{p}}_{1}^{1}}^{{\text{p}}_{1}^{1}}}\end{array}}$	${\text{p}}^{{\text{p}}_{\text{p}}^{\text{p}}}\!$
$65536\!$	${\begin{array}{lll}{\text{p}}_{1}^{16}&=&{\text{p}}_{1}^{{\text{p}}_{1}^{4}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{2}}}\\[12pt]&=&{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{{\text{p}}_{1}^{1}}}}\end{array}}$	${\text{p}}^{{\text{p}}^{{\text{p}}^{\text{p}}}}\!$

ASCII

 Example

    * k | natural numbers n such that |riff(n)| = k
    * 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 (x = root):
    * .................o.......o..o.......o
    * .................|.......^..|.......^
    * .................v.......|..v.......|
    * ...........o..o..o....o..o..o..o.o..o
    * ...........|..^..|....|..|..^..|.^..^
    * ...........v..|..v....v..v..|..v/...|
    * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
    * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;

A062537

Nodes in riff (rooted index-functional forest) for n.

Wiki + TeX + JPEG

$a(n)={\text{Number of Nodes in the Riff of}}~n$
$1\!$ $a(1)~=~0$	${\text{p}}\!$ $a(2)~=~1$	${\text{p}}_{\text{p}}\!$ $a(3)~=~2$	${\text{p}}^{\text{p}}\!$ $a(4)~=~2$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(5)~=~3$
${\text{p}}{\text{p}}_{\text{p}}\!$ $a(6)~=~3$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(7)~=~3$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(8)~=~3$	${\text{p}}_{\text{p}}^{\text{p}}\!$ $a(9)~=~3$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(10)~=~4$
${\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(11)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}\!$ $a(12)~=~4$	${\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(13)~=~4$	${\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(14)~=~4$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(15)~=~5$
${\text{p}}^{{\text{p}}^{\text{p}}}\!$ $a(16)~=~3$	${\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(17)~=~4$	${\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!$ $a(18)~=~4$	${\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(19)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(20)~=~5$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(21)~=~5$	${\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(22)~=~5$	${\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!$ $a(23)~=~4$	${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{\text{p}}\!$ $a(24)~=~5$	${\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!$ $a(25)~=~4$
${\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(26)~=~5$	${\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(27)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(28)~=~5$	${\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(29)~=~5$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(30)~=~6$
${\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}}\!$ $a(31)~=~5$	${\text{p}}^{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(32)~=~4$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(33)~=~6$	${\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(34)~=~5$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(35)~=~6$
${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!$ $a(36)~=~5$	${\text{p}}_{{\text{p}}^{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(37)~=~5$	${\text{p}}{\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(38)~=~5$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(39)~=~6$	${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(40)~=~6$
${\text{p}}_{{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}}\!$ $a(41)~=~5$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(42)~=~6$	${\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(43)~=~5$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(44)~=~6$	${\text{p}}_{\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(45)~=~6$
${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!$ $a(46)~=~5$	${\text{p}}_{{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(47)~=~6$	${\text{p}}^{{\text{p}}^{\text{p}}}{\text{p}}_{\text{p}}\!$ $a(48)~=~5$	${\text{p}}_{{\text{p}}^{\text{p}}}^{\text{p}}\!$ $a(49)~=~4$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!$ $a(50)~=~5$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(51)~=~6$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(52)~=~6$	${\text{p}}_{{\text{p}}^{{\text{p}}^{\text{p}}}}\!$ $a(53)~=~4$	${\text{p}}{\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(54)~=~5$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(55)~=~7$
${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(56)~=~6$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(57)~=~6$	${\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(58)~=~6$	${\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}}\!$ $a(59)~=~5$	${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(60)~=~7$

A062860

Smallest j with n nodes in its riff (rooted index-functional forest).

Wiki + TeX + JPEG

$a(n)={\text{Least Integer}}~j~{\text{with}}~n~{\text{Nodes in Its Riff}}$
$1\!$ $a(0)~=~1$	${\text{p}}\!$ $a(1)~=~2$	${\text{p}}_{\text{p}}\!$ $a(2)~=~3$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(3)~=~5$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(4)~=~10$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(5)~=~15$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(6)~=~30$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(7)~=~55$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(8)~=~105$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(9)~=~165$

A106177

Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function.

Primal Codes of Finite Partial Functions on Positive Integers

${\begin{array}{rcl}1&=&\varnothing \\2&=&1\!:\!1\\3&=&2\!:\!1\\4&=&1\!:\!2\\5&=&3\!:\!1\\6&=&1\!:\!1~~2\!:\!1\\7&=&4\!:\!1\\8&=&1\!:\!3\\9&=&2\!:\!2\\10&=&1\!:\!1~~3\!:\!1\\11&=&5\!:\!1\\12&=&1\!:\!2~~2\!:\!1\\13&=&6\!:\!1\\14&=&1\!:\!1~~4\!:\!1\\15&=&2\!:\!1~~3\!:\!1\\16&=&1\!:\!4\\17&=&7\!:\!1\\18&=&1\!:\!1~~2\!:\!2\\19&=&8\!:\!1\\20&=&1\!:\!2~~3\!:\!1\end{array}}$

Wiki Table

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

Wiki + TeX

Smallmatrix

${\begin{smallmatrix}&&&&&&&&&{\color {red}1}&&{\color {red}1}\\&&&&&&&&{\color {red}2}&&1&&{\color {red}2}\\&&&&&&&{\color {red}3}&&1&&1&&{\color {red}3}\\&&&&&&{\color {red}4}&&1&&2&&1&&{\color {red}4}\\&&&&&{\color {red}5}&&1&&3&&1&&1&&{\color {red}5}\\&&&&{\color {red}6}&&1&&1&&1&&4&&1&&{\color {red}6}\\&&&{\color {red}7}&&1&&5&&2&&9&&1&&1&&{\color {red}7}\\&&{\color {red}8}&&1&&6&&1&&1&&1&&2&&1&&{\color {red}8}\\&{\color {red}9}&&1&&7&&1&&25&&1&&3&&1&&1&&{\color {red}9}\\{\color {red}10}&&1&&1&&1&&36&&1&&2&&1&&8&&1&&{\color {red}10}\end{smallmatrix}}$

Array

${\begin{array}{*{21}{c}}&&&&&&&&&{\color {red}1}&&{\color {red}1}\\&&&&&&&&{\color {red}2}&&1&&{\color {red}2}\\&&&&&&&{\color {red}3}&&1&&1&&{\color {red}3}\\&&&&&&{\color {red}4}&&1&&2&&1&&{\color {red}4}\\&&&&&{\color {red}5}&&1&&3&&1&&1&&{\color {red}5}\\&&&&{\color {red}6}&&1&&1&&1&&4&&1&&{\color {red}6}\\&&&{\color {red}7}&&1&&5&&2&&9&&1&&1&&{\color {red}7}\\&&{\color {red}8}&&1&&6&&1&&1&&1&&2&&1&&{\color {red}8}\\&{\color {red}9}&&1&&7&&1&&25&&1&&3&&1&&1&&{\color {red}9}\\{\color {red}10}&&1&&1&&1&&36&&1&&2&&1&&8&&1&&{\color {red}10}\end{array}}$

Matrix

${\begin{matrix}n\circ m\\1~/~\backslash ~1\\2~/~1~\backslash ~2\\3~/~1\cdot 1~\backslash ~3\\4~/~1\cdot 2\cdot 1~\backslash ~4\\5~/~1\cdot 3\cdot 1\cdot 1~\backslash ~5\\6~/~1\cdot 1\cdot 1\cdot 4\cdot 1~\backslash ~6\\7~/~1\cdot 5\cdot 2\cdot 9\cdot 1\cdot 1~\backslash ~7\\8~/~1\cdot 6\cdot 1\cdot 1\cdot 1\cdot 2\cdot 1~\backslash ~8\\9~/~1\cdot 7\cdot 1\cdot 25\cdot 1\cdot 3\cdot 1\cdot 1~\backslash ~9\\10~/~1\cdot 1\cdot 1\cdot 36\cdot 1\cdot 2\cdot 1\cdot 8\cdot 1~\backslash ~10\end{matrix}}$

ASCII

 Example

    *                      n o m
    *                       \ /
    *                      1 . 1
    *                     \ / \ /
    *                    2 . 1 . 2
    *                   \ / \ / \ /
    *                  3 . 1 . 1 . 3
    *                 \ / \ / \ / \ /
    *                4 . 1 . 2 . 1 . 4
    *               \ / \ / \ / \ / \ /
    *              5 . 1 . 3 . 1 . 1 . 5
    *             \ / \ / \ / \ / \ / \ /
    *            6 . 1 . 1 . 1 . 4 . 1 . 6
    *           \ / \ / \ / \ / \ / \ / \ /
    *          7 . 1 . 5 . 2 . 9 . 1 . 1 . 7
    *         \ / \ / \ / \ / \ / \ / \ / \ /
    *        8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8
    *       \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9
    *     \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *   10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10
    *
    * Primal codes of finite partial functions on positive integers:
    * 1 = { }
    * 2 = 1:1
    * 3 = 2:1
    * 4 = 1:2
    * 5 = 3:1
    * 6 = 1:1 2:1
    * 7 = 4:1
    * 8 = 1:3
    * 9 = 2:2
    * 10 = 1:1 3:1
    * 11 = 5:1
    * 12 = 1:2 2:1
    * 13 = 6:1
    * 14 = 1:1 4:1
    * 15 = 2:1 3:1
    * 16 = 1:4
    * 17 = 7:1
    * 18 = 1:1 2:2
    * 19 = 8:1
    * 20 = 1:2 3:1

A106178

Functional composition table for "n o m" = "n composed with m", where n and m are the "primal codes" of finite partial functions on the positive integers and 1 is the code for the empty function, but omitting the trivial values of 1 at the margins of the table.

Wiki Table

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

TeX Smallmatrix

${\begin{smallmatrix}&&&&&&&&&&&&&&&{\color {red}1}&&{\color {red}1}\\&&&&&&&&&&&&&&{\color {red}2}&&\cdot &&{\color {red}2}\\&&&&&&&&&&&&&{\color {red}3}&&\cdot &&\cdot &&{\color {red}3}\\&&&&&&&&&&&&{\color {red}4}&&\cdot &&2&&\cdot &&{\color {red}4}\\&&&&&&&&&&&{\color {red}5}&&\cdot &&3&&1&&\cdot &&{\color {red}5}\\&&&&&&&&&&{\color {red}6}&&\cdot &&1&&1&&4&&\cdot &&{\color {red}6}\\&&&&&&&&&{\color {red}7}&&\cdot &&5&&2&&9&&1&&\cdot &&{\color {red}7}\\&&&&&&&&{\color {red}8}&&\cdot &&6&&1&&1&&1&&2&&\cdot &&{\color {red}8}\\&&&&&&&{\color {red}9}&&\cdot &&7&&1&&25&&1&&3&&1&&\cdot &&{\color {red}9}\\&&&&&&{\color {red}10}&&\cdot &&1&&1&&36&&1&&2&&1&&8&&\cdot &&{\color {red}10}\\&&&&&{\color {red}11}&&\cdot &&1&&1&&49&&1&&5&&1&&27&&1&&\cdot &&{\color {red}11}\\&&&&{\color {red}12}&&\cdot &&10&&3&&1&&1&&6&&1&&1&&1&&2&&\cdot &&{\color {red}12}\\&&&{\color {red}13}&&\cdot &&11&&1&&1&&2&&7&&1&&125&&4&&3&&1&&\cdot &&{\color {red}13}\\&&{\color {red}14}&&\cdot &&3&&1&&100&&1&&1&&1&&216&&1&&1&&1&&4&&\cdot &&{\color {red}14}\\&{\color {red}15}&&\cdot &&13&&2&&121&&1&&3&&1&&343&&1&&5&&1&&9&&1&&\cdot &&{\color {red}15}\\{\color {red}16}&&\cdot &&14&&1&&9&&1&&10&&1&&1&&1&&6&&1&&2&&1&&2&&\cdot &&{\color {red}16}\end{smallmatrix}}$

ASCII

 Example

    *                                   n o m
    *                                    \ /
    *                                   1 . 1
    *                                  \ / \ /
    *                                 2 .   . 2
    *                                \ / \ / \ /
    *                               3 .   .   . 3
    *                              \ / \ / \ / \ /
    *                             4 .   . 2 .   . 4
    *                            \ / \ / \ / \ / \ /
    *                           5 .   . 3 . 1 .   . 5
    *                          \ / \ / \ / \ / \ / \ /
    *                         6 .   . 1 . 1 . 4 .   . 6
    *                        \ / \ / \ / \ / \ / \ / \ /
    *                       7 .   . 5 . 2 . 9 . 1 .   . 7
    *                      \ / \ / \ / \ / \ / \ / \ / \ /
    *                     8 .   . 6 . 1 . 1 . 1 . 2 .   . 8
    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                   9 .   . 7 . 1 . 25. 1 . 3 . 1 .   . 9
    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                10 .   . 1 . 1 . 36. 1 . 2 . 1 . 8 .   . 10
    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *              11 .   . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 .   . 11
    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *            12 .   . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 .   . 12
    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *          13 .   . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 .   . 13
    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *        14 .   . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 .   . 14
    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      15 .   . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 .   . 15
    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *    16 .   . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 .   . 16

A108352

a(n) = primal code characteristic of n, which is the least positive integer, if any, such that (n o)^k = 1, otherwise equal to 0. Here "o" denotes the primal composition operator, as illustrated in A106177 and A108371, and (n o)^k = n o … o n, with k occurrences of n.

Links

Jon Awbrey, Primal Code Characteristic, n = 1 to 1000
Jon Awbrey, Primal Code Characteristic, n = 1001 to 2000
Jon Awbrey, Primal Code Characteristic, n = 2001 to 3000

TeX Array

${\begin{array}{*{10}{l}}a(1)&=&1&{\text{because}}&(\circ ~1)^{1}&=&(\circ ~\varnothing )^{1}&=&1.\\a(2)&=&0&{\text{because}}&(\circ ~2)^{k}&=&(\circ ~1\!:\!1)^{k}&=&2,&{\text{for all}}~k>0.\\a(3)&=&2&{\text{because}}&(\circ ~3)^{2}&=&(\circ ~2\!:\!1)^{2}&=&1.\\a(4)&=&2&{\text{because}}&(\circ ~4)^{2}&=&(\circ ~1\!:\!2)^{2}&=&1.\\a(5)&=&2&{\text{because}}&(\circ ~5)^{2}&=&(\circ ~3\!:\!1)^{2}&=&1.\\a(6)&=&0&{\text{because}}&(\circ ~6)^{k}&=&(\circ ~1\!:\!1~~2\!:\!1)^{k}&=&6,&{\text{for all}}~k>0.\\a(7)&=&2&{\text{because}}&(\circ ~7)^{2}&=&(\circ ~4\!:\!1)^{1}&=&1.\\a(8)&=&2&{\text{because}}&(\circ ~8)^{2}&=&(\circ ~1\!:\!3)^{1}&=&1.\\a(9)&=&0&{\text{because}}&(\circ ~9)^{k}&=&(\circ ~2\!:\!2)^{k}&=&9,&{\text{for all}}~k>0.\\a(10)&=&0&{\text{because}}&(\circ ~10)^{k}&=&(\circ ~1\!:\!1~~3\!:\!1)^{k}&=&10,&{\text{for all}}~k>0.\end{array}}$

ASCII

Example

    * a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1.
    * a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k.
    * a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1.
    * a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1.
    * a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1.
    * a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k.
    * a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1.
    * a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1.
    * a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k.
    * a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k.
    * Detail of calculation for compositional powers of 12:
    * (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18
    * (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12
    * Detail of calculation for compositional powers of 20:
    * (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25
    * (20 o)^3 = (3:2) o (1:2 3:1) = 1

A108353

For each nonnegative integer n, a(n) is the smallest positive integer j whose primal code characteristic is n, that is, the smallest j such that A108352(j) = n.

TeX Array

${\begin{array}{rcl||l}2&=&1\!:\!1&1\mapsto 1~{\text{(infinite loop)}}\\1&=&\varnothing &1\\3&=&2\!:\!1&2\mapsto 1\\20&=&1\!:\!2~~3\!:\!1&3\mapsto 1\mapsto 2\\756&=&1\!:\!2~~2\!:\!3~~4\!:\!1&4\mapsto 1\mapsto 2\mapsto 3\\178200&=&1\!:\!3~~2\!:\!4~~3\!:\!2~~5\!:\!1&5\mapsto 1\mapsto 3\mapsto 2\mapsto 4\end{array}}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Functions and Functional Digraphs for a(0) to a(5)
    *       2 = 1:1             || 1 -> 1 (infinite loop)
    *       1 = { }             || 1
    *       3 = 2:1             || 2 -> 1
    *      20 = 1:2 3:1         || 3 -> 1 -> 2
    *     756 = 1:2 2:3 4:1     || 4 -> 1 -> 2 -> 3
    *  178200 = 1:3 2:4 3:2 5:1 || 5 -> 1 -> 3 -> 2 -> 4

A108370

Numbers whose primal code characteristic = 0, that is, positive n for which A108352(n) = 0.

A108371

Table of primal compositional powers (n o)^k, where "o" denotes the primal composition operator, as illustrated in sequence A106177, and where (n o)^k = n o … o n, with k occurrences of n.

Wiki Table

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

ASCII

 Example

    * Table: T(n,k) = (n o)^k
    *                                  T(n,k)
    *                                    \ /
    *                                   1 . 1
    *                                  \ / \ /
    *                                 2 . 1 . 2
    *                                \ / \ / \ /
    *                               3 . 2 . 1 . 3
    *                              \ / \ / \ / \ /
    *                             4 . 3 . 2 . 1 . 4
    *                            \ / \ / \ / \ / \ /
    *                           5 . 4 . 1 . 2 . 1 . 5
    *                          \ / \ / \ / \ / \ / \ /
    *                         6 . 5 . 1 . 1 . 2 . 1 . 6
    *                        \ / \ / \ / \ / \ / \ / \ /
    *                       7 . 6 . 1 . 1 . 1 . 2 . 1 . 7
    *                      \ / \ / \ / \ / \ / \ / \ / \ /
    *                     8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8
    *                    \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                   9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9
    *                  \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *                10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10
    *                \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *              11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11
    *              \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *            12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12
    *            \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *          13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13
    *          \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *        14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14
    *        \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *      15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15
    *      \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
    *    16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16

A108372

Numbers whose primal code characteristic = 2, that is, positive n for which A108352(n) = 2.

A108373

Numbers whose primal code characteristic = 3, that is, positive n for which A108352(n) = 3.

A108374

Numbers whose primal code characteristic = 4, that is, positive n for which A108352(n) = 4.

TeX Array

${\begin{array}{rcccc||l}756&=&1\!:\!2&2\!:\!3&4\!:\!1&4\mapsto 1\mapsto 2\mapsto 3\\1176&=&1\!:\!3&2\!:\!1&4\!:\!2&4\mapsto 2\mapsto 1\mapsto 3\\1188&=&1\!:\!2&2\!:\!3&5\!:\!1&5\mapsto 1\mapsto 2\mapsto 3\\1200&=&1\!:\!4&2\!:\!1&3\!:\!2&3\mapsto 2\mapsto 1\mapsto 4\\1400&=&1\!:\!3&3\!:\!2&4\!:\!1&4\mapsto 1\mapsto 3\mapsto 2\\1404&=&1\!:\!2&2\!:\!3&6\!:\!1&6\mapsto 1\mapsto 2\mapsto 3\\1620&=&1\!:\!2&2\!:\!4&3\!:\!1&3\mapsto 1\mapsto 2\mapsto 4\\1836&=&1\!:\!2&2\!:\!3&7\!:\!1&7\mapsto 1\mapsto 2\mapsto 3\\2052&=&1\!:\!2&2\!:\!3&8\!:\!1&8\mapsto 1\mapsto 2\mapsto 3\\2160&=&1\!:\!4&2\!:\!3&3\!:\!1&2\mapsto 3\mapsto 1\mapsto 4\\2200&=&1\!:\!3&3\!:\!2&5\!:\!1&5\mapsto 1\mapsto 3\mapsto 2\\2400&=&1\!:\!5&2\!:\!1&3\!:\!2&3\mapsto 2\mapsto 1\mapsto 5\\2484&=&1\!:\!2&2\!:\!3&9\!:\!1&9\mapsto 1\mapsto 2\mapsto 3\\2600&=&1\!:\!3&3\!:\!2&6\!:\!1&6\mapsto 1\mapsto 3\mapsto 2\\2904&=&1\!:\!3&2\!:\!1&5\!:\!2&5\mapsto 2\mapsto 1\mapsto 3\end{array}}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Functions and Functional Digraphs for a(1) to a(15)
    * 0756 = 1:2 2:3 4:1 || 4 -> 1 -> 2 -> 3
    * 1176 = 1:3 2:1 4:2 || 4 -> 2 -> 1 -> 3
    * 1188 = 1:2 2:3 5:1 || 5 -> 1 -> 2 -> 3
    * 1200 = 1:4 2:1 3:2 || 3 -> 2 -> 1 -> 4
    * 1400 = 1:3 3:2 4:1 || 4 -> 1 -> 3 -> 2
    * 1404 = 1:2 2:3 6:1 || 6 -> 1 -> 2 -> 3
    * 1620 = 1:2 2:4 3:1 || 3 -> 1 -> 2 -> 4
    * 1836 = 1:2 2:3 7:1 || 7 -> 1 -> 2 -> 3
    * 2052 = 1:2 2:3 8:1 || 8 -> 1 -> 2 -> 3
    * 2160 = 1:4 2:3 3:1 || 2 -> 3 -> 1 -> 4
    * 2200 = 1:3 3:2 5:1 || 5 -> 1 -> 3 -> 2
    * 2400 = 1:5 2:1 3:2 || 3 -> 2 -> 1 -> 5
    * 2484 = 1:2 2:3 9:1 || 9 -> 1 -> 2 -> 3
    * 2600 = 1:3 3:2 6:1 || 6 -> 1 -> 3 -> 2
    * 2904 = 1:3 2:1 5:2 || 5 -> 2 -> 1 -> 3

A109297

Primal codes of finite permutations on positive integers.

TeX Array

${\begin{array}{rclll}1&=&\varnothing \\2&=&1\!:\!1\\9&=&2\!:\!2\\12&=&1\!:\!2&2\!:\!1\\18&=&1\!:\!1&2\!:\!2\\40&=&1\!:\!3&3\!:\!1\\112&=&1\!:\!4&4\!:\!1\\125&=&3\!:\!3\\250&=&1\!:\!1&3\!:\!3\\352&=&1\!:\!5&5\!:\!1\\360&=&1\!:\!3&2\!:\!2&3\!:\!1\\540&=&1\!:\!2&2\!:\!3&3\!:\!1\\600&=&1\!:\!3&2\!:\!1&3\!:\!2\\675&=&2\!:\!3&3\!:\!2\\832&=&1\!:\!6&6\!:\!1\\1008&=&1\!:\!4&2\!:\!2&4\!:\!1\\1125&=&2\!:\!2&3\!:\!3\\1350&=&1\!:\!1&2\!:\!3&3\!:\!2\\1500&=&1\!:\!2&2\!:\!1&3\!:\!3\\2176&=&1\!:\!7&7\!:\!1\\2250&=&1\!:\!1&2\!:\!2&3\!:\!3\end{array}}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table:
    * Primal Codes of Finite Permutations on Positive Integers
    *       1 = { }
    *       2 = 1:1
    *       9 = 2:2
    *      12 = 1:2 2:1
    *      18 = 1:1 2:2
    *      40 = 1:3 3:1
    *     112 = 1:4 4:1
    *     125 = 3:3
    *     250 = 1:1 3:3
    *     352 = 1:5 5:1
    *     360 = 1:3 2:2 3:1
    *     540 = 1:2 2:3 3:1
    *     600 = 1:3 2:1 3:2
    *     675 = 2:3 3:2
    *     832 = 1:6 6:1
    *    1008 = 1:4 2:2 4:1
    *    1125 = 2:2 3:3
    *    1350 = 1:1 2:3 3:2
    *    1500 = 1:2 2:1 3:3
    *    2176 = 1:7 7:1
    *    2250 = 1:1 2:2 3:3

A109298

Primal codes of finite idempotent functions on positive integers.

TeX Array

${\begin{array}{rcllll}1&=&\varnothing \\2&=&1\!:\!1\\9&=&&2\!:\!2\\18&=&1\!:\!1&2\!:\!2\\125&=&&&3\!:\!3\\250&=&1\!:\!1&&3\!:\!3\\1125&=&&2\!:\!2&3\!:\!3\\2250&=&1\!:\!1&2\!:\!2&3\!:\!3\\2401&=&&&&4\!:\!4\\4802&=&1\!:\!1&&&4\!:\!4\\21609&=&&2\!:\!2&&4\!:\!4\\43218&=&1\!:\!1&2\!:\!2&&4\!:\!4\\300125&=&&&3\!:\!3&4\!:\!4\\600250&=&1\!:\!1&&3\!:\!3&4\!:\!4\\2701125&=&&2\!:\!2&3\!:\!3&4\!:\!4\\5402250&=&1\!:\!1&2\!:\!2&3\!:\!3&4\!:\!4\end{array}}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have the following table of examples:
    * Primal Codes of Finite Idempotent Functions on Positive Integers
    *       1 = { }
    *       2 = 1:1
    *       9 =     2:2
    *      18 = 1:1 2:2
    *     125 =         3:3
    *     250 = 1:1     3:3
    *    1125 =     2:2 3:3
    *    2250 = 1:1 2:2 3:3
    *    2401 =             4:4
    *    4802 = 1:1         4:4
    *   21609 =     2:2     4:4
    *   43218 = 1:1 2:2     4:4
    *  300125 =         3:3 4:4
    *  600250 = 1:1     3:3 4:4
    * 2701125 =     2:2 3:3 4:4
    * 5402250 = 1:1 2:2 3:3 4:4

A109299

Primal codes of canonical finite permutations on positive integers.

TeX Array

${\begin{array}{rcllll}1&=&\varnothing \\2&=&1\!:\!1\\12&=&1\!:\!2&2\!:\!1\\18&=&1\!:\!1&2\!:\!2\\360&=&1\!:\!3&2\!:\!2&3\!:\!1\\540&=&1\!:\!2&2\!:\!3&3\!:\!1\\600&=&1\!:\!3&2\!:\!1&3\!:\!2\\1350&=&1\!:\!1&2\!:\!3&3\!:\!2\\1500&=&1\!:\!2&2\!:\!1&3\!:\!3\\2250&=&1\!:\!1&2\!:\!2&3\!:\!3\\75600&=&1\!:\!4&2\!:\!3&3\!:\!2&4\!:\!1\\105840&=&1\!:\!4&2\!:\!3&3\!:\!1&4\!:\!2\\113400&=&1\!:\!3&2\!:\!4&3\!:\!2&4\!:\!1\\126000&=&1\!:\!4&2\!:\!2&3\!:\!3&4\!:\!1\\158760&=&1\!:\!3&2\!:\!4&3\!:\!1&4\!:\!2\\246960&=&1\!:\!4&2\!:\!2&3\!:\!1&4\!:\!3\\283500&=&1\!:\!2&2\!:\!4&3\!:\!3&4\!:\!1\\294000&=&1\!:\!4&2\!:\!1&3\!:\!3&4\!:\!2\\315000&=&1\!:\!3&2\!:\!2&3\!:\!4&4\!:\!1\\411600&=&1\!:\!4&2\!:\!1&3\!:\!2&4\!:\!3\\472500&=&1\!:\!2&2\!:\!3&3\!:\!4&4\!:\!1\\555660&=&1\!:\!2&2\!:\!4&3\!:\!1&4\!:\!3\\735000&=&1\!:\!3&2\!:\!1&3\!:\!4&4\!:\!2\\864360&=&1\!:\!3&2\!:\!2&3\!:\!1&4\!:\!4\\992250&=&1\!:\!1&2\!:\!4&3\!:\!3&4\!:\!2\\1296540&=&1\!:\!2&2\!:\!3&3\!:\!1&4\!:\!4\\1389150&=&1\!:\!1&2\!:\!4&3\!:\!2&4\!:\!3\\1440600&=&1\!:\!3&2\!:\!1&3\!:\!2&4\!:\!4\\1653750&=&1\!:\!1&2\!:\!3&3\!:\!4&4\!:\!2\\2572500&=&1\!:\!2&2\!:\!1&3\!:\!4&4\!:\!3\\3241350&=&1\!:\!1&2\!:\!3&3\!:\!2&4\!:\!4\\3601500&=&1\!:\!2&2\!:\!1&3\!:\!3&4\!:\!4\\3858750&=&1\!:\!1&2\!:\!2&3\!:\!4&4\!:\!3\\5402250&=&1\!:\!1&2\!:\!2&3\!:\!3&4\!:\!4\end{array}}$

ASCII

 Example

    * Writing (prime(i))^j as i:j, we have this table:
    * Primal Codes of Canonical Finite Permutations
    *       1 = { }
    *       2 = 1:1
    *      12 = 1:2 2:1
    *      18 = 1:1 2:2
    *     360 = 1:3 2:2 3:1
    *     540 = 1:2 2:3 3:1
    *     600 = 1:3 2:1 3:2
    *    1350 = 1:1 2:3 3:2
    *    1500 = 1:2 2:1 3:3
    *    2250 = 1:1 2:2 3:3
    *   75600 = 1:4 2:3 3:2 4:1
    *  105840 = 1:4 2:3 3:1 4:2
    *  113400 = 1:3 2:4 3:2 4:1
    *  126000 = 1:4 2:2 3:3 4:1
    *  158760 = 1:3 2:4 3:1 4:2
    *  246960 = 1:4 2:2 3:1 4:3
    *  283500 = 1:2 2:4 3:3 4:1
    *  294000 = 1:4 2:1 3:3 4:2
    *  315000 = 1:3 2:2 3:4 4:1
    *  411600 = 1:4 2:1 3:2 4:3
    *  472500 = 1:2 2:3 3:4 4:1
    *  555660 = 1:2 2:4 3:1 4:3
    *  735000 = 1:3 2:1 3:4 4:2
    *  864360 = 1:3 2:2 3:1 4:4
    *  992250 = 1:1 2:4 3:3 4:2
    * 1296540 = 1:2 2:3 3:1 4:4
    * 1389150 = 1:1 2:4 3:2 4:3
    * 1440600 = 1:3 2:1 3:2 4:4
    * 1653750 = 1:1 2:3 3:4 4:2
    * 2572500 = 1:2 2:1 3:4 4:3
    * 3241350 = 1:1 2:3 3:2 4:4
    * 3601500 = 1:2 2:1 3:3 4:4
    * 3858750 = 1:1 2:2 3:4 4:3
    * 5402250 = 1:1 2:2 3:3 4:4

A109300

a(n) = number of positive integers whose rote height in gammas is n.

JPEG

${\text{Table 1.}}~~{\text{Rotes and Primal Functions for Positive Integers of Rote Height 2}}\!$
${\begin{array}{l}2\!:\!1\\3\end{array}}$	${\begin{array}{l}1\!:\!2\\4\end{array}}$	${\begin{array}{l}1\!:\!1~~2\!:\!1\\6\end{array}}$	${\begin{array}{l}2\!:\!2\\9\end{array}}$	${\begin{array}{l}1\!:\!2~~2\!:\!1\\12\end{array}}$	${\begin{array}{l}1\!:\!1~~2\!:\!2\\18\end{array}}$	${\begin{array}{l}1\!:\!2~~2\!:\!2\\36\end{array}}$

ASCII

 Example

    * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2
    *                                                                          
    * o-o     o-o       o-o   o-o o-o     o-o o-o       o-o o-o     o-o o-o o-o
    * |       |         |     |   |       |   |         |   |       |   |   |  
    * o-o   o-o     o-o o-o   o---o     o-o   o-o   o-o o---o     o-o   o---o  
    * |     |       |   |     |         |     |     |   |         |     |      
    * O     O       O===O     O         O=====O     O===O         O=====O      
    *                                                                          
    * 2:1   1:2     1:1 2:1   2:2       1:2 2:1     1:1 2:2       1:2 2:2      
    *                                                                          
    * 3     4       6         9         12          18            36           
    *

A109301

a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.

Example

802701=9\cdot 89189={\text{p}}_{2}^{2}{\text{p}}_{8638}^{1}

{\text{Writing}}~(\operatorname {prime} (i))^{j}~{\text{as}}~i\!:\!j,~{\text{we have:}}

{\begin{array}{lllll}802701&=&9\cdot 89189&=&2\!:\!2~~8638\!:\!1\\8638&=&2\cdot 7\cdot 617&=&1\!:\!1~~4\!:\!1~~113\!:\!1\\113&&&=&30\!:\!1\\30&=&2\cdot 3\cdot 5&=&1\!:\!1~~2\!:\!1~~3\!:\!1\\4&&&=&1\!:\!2\\3&&&=&2\!:\!1\\2&&&=&1\!:\!1\end{array}}

{\text{So the rote of 802701 is the following graph:}}\!

{\text{By inspection, the rote height of 802701 is 6.}}\!

JPEG

$a(n)={\text{Rote Height of}}~n$
$1\!$ $a(1)~=~0$	${\text{p}}\!$ $a(2)~=~1$	${\text{p}}_{\text{p}}\!$ $a(3)~=~2$	${\text{p}}^{\text{p}}\!$ $a(4)~=~2$	${\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(5)~=~3$
${\text{p}}{\text{p}}_{\text{p}}\!$ $a(6)~=~2$	${\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(7)~=~3$	${\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(8)~=~3$	${\text{p}}_{\text{p}}^{\text{p}}\!$ $a(9)~=~2$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(10)~=~3$
${\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(11)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}\!$ $a(12)~=~2$	${\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(13)~=~3$	${\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(14)~=~3$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(15)~=~3$
${\text{p}}^{{\text{p}}^{\text{p}}}\!$ $a(16)~=~3$	${\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(17)~=~4$	${\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!$ $a(18)~=~2$	${\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(19)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(20)~=~3$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(21)~=~3$	${\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(22)~=~4$	${\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!$ $a(23)~=~3$	${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{\text{p}}\!$ $a(24)~=~3$	${\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!$ $a(25)~=~3$
${\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(26)~=~3$	${\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(27)~=~3$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(28)~=~3$	${\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(29)~=~4$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(30)~=~3$
${\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}}\!$ $a(31)~=~5$	${\text{p}}^{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(32)~=~4$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(33)~=~4$	${\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(34)~=~4$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(35)~=~3$
${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}^{\text{p}}\!$ $a(36)~=~2$	${\text{p}}_{{\text{p}}^{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(37)~=~3$	${\text{p}}{\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(38)~=~4$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(39)~=~3$	${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(40)~=~3$
${\text{p}}_{{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}}\!$ $a(41)~=~4$	${\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(42)~=~3$	${\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(43)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(44)~=~4$	${\text{p}}_{\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(45)~=~3$
${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}^{\text{p}}}\!$ $a(46)~=~3$	${\text{p}}_{{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(47)~=~4$	${\text{p}}^{{\text{p}}^{\text{p}}}{\text{p}}_{\text{p}}\!$ $a(48)~=~3$	${\text{p}}_{{\text{p}}^{\text{p}}}^{\text{p}}\!$ $a(49)~=~3$	${\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}^{\text{p}}\!$ $a(50)~=~3$
${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}\!$ $a(51)~=~4$	${\text{p}}^{\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{\text{p}}}\!$ $a(52)~=~3$	${\text{p}}_{{\text{p}}^{{\text{p}}^{\text{p}}}}\!$ $a(53)~=~4$	${\text{p}}{\text{p}}_{\text{p}}^{{\text{p}}_{\text{p}}}\!$ $a(54)~=~3$	${\text{p}}_{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(55)~=~4$
${\text{p}}^{{\text{p}}_{\text{p}}}{\text{p}}_{{\text{p}}^{\text{p}}}\!$ $a(56)~=~3$	${\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}^{{\text{p}}_{\text{p}}}}\!$ $a(57)~=~4$	${\text{p}}{\text{p}}_{{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}}\!$ $a(58)~=~4$	${\text{p}}_{{\text{p}}_{{\text{p}}_{{\text{p}}^{\text{p}}}}}\!$ $a(59)~=~5$	${\text{p}}^{\text{p}}{\text{p}}_{\text{p}}{\text{p}}_{{\text{p}}_{\text{p}}}\!$ $a(60)~=~3$

ASCII

 Comment

    * Table of Rotes and Primal Functions for Positive Integers from 1 to 40
    *                                                                        
    *                                                         o-o            
    *                                                         |              
    *                             o-o             o-o         o-o            
    *                             |               |           |              
    *               o-o           o-o           o-o           o-o            
    *               |             |             |             |              
    * O             O             O             O             O              
    *                                                                        
    * { }           1:1           2:1           1:2           3:1            
    *                                                                        
    * 1             2             3             4             5              
    *                                                                        
    *                                                                        
    *                 o-o           o-o                           o-o        
    *                 |             |                             |          
    *     o-o       o-o             o-o         o-o o-o           o-o        
    *     |         |               |           |   |             |          
    * o-o o-o       o-o           o-o           o---o         o-o o-o        
    * |   |         |             |             |             |   |          
    * O===O         O             O             O             O===O          
    *                                                                        
    * 1:1 2:1       4:1           1:3           2:2           1:1 3:1        
    *                                                                        
    * 6             7             8             9             10             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o                             o-o             o-o         o-o        
    * |                               |               |           |          
    * o-o             o-o o-o     o-o o-o           o-o       o-o o-o        
    * |               |   |       |   |             |         |   |          
    * o-o           o-o   o-o     o===o-o       o-o o-o       o-o o-o        
    * |             |     |       |             |   |         |   |          
    * O             O=====O       O             O===O         O===O          
    *                                                                        
    * 5:1           1:2 2:1       6:1           1:1 4:1       2:1 3:1        
    *                                                                        
    * 11            12            13            14            15             
    *                                                                        
    *                                                                        
    *                 o-o                         o-o                        
    *                 |                           |                          
    *     o-o       o-o                           o-o               o-o      
    *     |         |                             |                 |        
    *   o-o         o-o               o-o o-o   o-o             o-o o-o      
    *   |           |                 |   |     |               |   |        
    * o-o           o-o           o-o o---o     o-o           o-o   o-o      
    * |             |             |   |         |             |     |        
    * O             O             O===O         O             O=====O        
    *                                                                        
    * 1:4           7:1           1:1 2:2       8:1           1:2 3:1        
    *                                                                        
    * 16            17            18            19            20             
    *                                                                        
    *                                                                        
    *                   o-o                                                  
    *                   |                                                    
    *       o-o         o-o       o-o o-o         o-o         o-o            
    *       |           |         |   |           |           |              
    * o-o o-o           o-o       o---o           o-o o-o     o-o o-o        
    * |   |             |         |               |   |       |   |          
    * o-o o-o       o-o o-o       o-o           o-o   o-o     o---o          
    * |   |         |   |         |             |     |       |              
    * O===O         O===O         O             O=====O       O              
    *                                                                        
    * 2:1 4:1       1:1 5:1       9:1           1:3 2:1       3:2            
    *                                                                        
    * 21            22            23            24            25             
    *                                                                        
    *                                                                        
    *                                               o-o                      
    *                                               |                        
    *         o-o       o-o               o-o       o-o               o-o    
    *         |         |                 |         |                 |      
    *     o-o o-o   o-o o-o         o-o o-o     o-o o-o           o-o o-o    
    *     |   |     |   |           |   |       |   |             |   |      
    * o-o o===o-o   o---o         o-o   o-o     o===o-o       o-o o-o o-o    
    * |   |         |             |     |       |             |   |   |      
    * O===O         O             O=====O       O             O===O===O      
    *                                                                        
    * 1:1 6:1       2:3           1:2 4:1       10:1          1:1 2:1 3:1    
    *                                                                        
    * 26            27            28            29            30             
    *                                                                        
    *                                                                        
    * o-o                                                                    
    * |                                                                      
    * o-o             o-o             o-o             o-o                    
    * |               |               |               |                      
    * o-o             o-o             o-o           o-o       o-o   o-o      
    * |               |               |             |         |     |        
    * o-o             o-o         o-o o-o           o-o       o-o o-o        
    * |               |           |   |             |         |   |          
    * o-o           o-o           o-o o-o       o-o o-o       o-o o-o        
    * |             |             |   |         |   |         |   |          
    * O             O             O===O         O===O         O===O          
    *                                                                        
    * 11:1          1:5           2:1 5:1       1:1 7:1       3:1 4:1        
    *                                                                        
    * 31            32            33            34            35             
    *                                                                        
    *                                                                        
    *                                   o-o                                  
    *                                   |                                    
    *                 o-o o-o           o-o             o-o     o-o o-o      
    *                 |   |             |               |       |   |        
    *   o-o o-o o-o o-o   o-o         o-o       o-o o-o o-o     o-o o-o      
    *   |   |   |   |     |           |         |   |   |       |   |        
    * o-o   o---o   o=====o-o     o-o o-o       o-o o===o-o   o-o   o-o      
    * |     |       |             |   |         |   |         |     |        
    * O=====O       O             O===O         O===O         O=====O        
    *                                                                        
    * 1:2 2:2       12:1          1:1 8:1       2:1 6:1       1:3 3:1        
    *                                                                        
    * 36            37            38            39            40             
    *                                                                        
    * In these Figures, "extended lines of identity" like o===o
    * indicate identified nodes and capital O is the root node.
    * The rote height in gammas is found by finding the number
    * of graphs of the following shape between the root and one
    * of the highest nodes of the tree:
    * o--o
    * |
    * o
    * A sequence like this, that can be regarded as a nonnegative integer
    * measure on positive integers, may have as many as 3 other sequences
    * associated with it. Given that the fiber of a function f at n is all
    * the domain elements that map to n, we always have the fiber minimum
    * or minimum inverse function and may also have the fiber cardinality
    * and the fiber maximum or maximum inverse function. For A109301, the
    * minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the
    * first positive integer whose rote height is n, the fiber cardinality
    * is A109300, giving the number of positive integers of rote height n,
    * while the maximum inverse, g(n) = max {k : A109301(k) = n}, giving
    * the last positive integer whose rote height is n, has the following
    * initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36,
    * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 =
    * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly
    * 7.840858554516122655953405327738 x 10^371.

 Example

    * Writing (prime(i))^j as i:j, we have:
    * 802701 = 2:2 8638:1
    * 8638 = 1:1 4:1 113:1
    * 113 = 30:1
    * 30 = 1:1 2:1 3:1
    * 4 = 1:2
    * 3 = 2:1
    * 2 = 1:1
    * 1 = { }
    * So rote(802701) is the graph:
    *                              
    *                           o-o
    *                           |  
    *                       o-o o-o
    *                       |   |  
    *               o-o o-o o-o o-o
    *               |   |   |   |  
    *             o-o   o===o===o-o
    *             |     |          
    * o-o o-o o-o o-o   o---------o
    * |   |   |   |     |          
    * o---o   o===o=====o---------o
    * |       |                    
    * O=======O                    
    *                              
    * Therefore rhig(802701) = 6.

A111788

Order of the domain D_n (n >= 0) in the inverse limit domain D_infinity.

A111789

First differences of (0, A111788), the sequence that begins with 0 and continues with the terms of A111788.

A111790

Partial sums of A111788.

A111791

Positive integers sorted by rote height, as measured by A109301.

TeX Array

${\begin{array}{l|l|r}h&S_{h}~=~\{m~:~\operatorname {rote~height} (m)~=~\operatorname {A109301} (m)~=~h\}&|S_{h}|\\\hline \hline 0&\{1\}&1\\\hline 1&\{2\}&1\\\hline 2&\{3,4,6,9,12,18,36\}&7\\\hline 3&\{5,7,8,10,13,14,15,16,20,21,23,24,25,26,27,28,30,\\&35,37,39,40,42,45,46,48,49,50,52,54,56,60,61,63,\\&64,65,69,70,72,74,75,78,80,81,84,90,91,92,98,100,\ldots \}&999999991\\\hline 4&\{11,17,19,22,29,32,33,34,38,41,43,44,47,51,53,55,\\&57,58,66,68,71,73,76,77,82,83,85,86,87,88,89,94,\\&95,96,97,99,\ldots \}&\operatorname {A109300} (4)\\\hline 5&\{31,59,62,67,79,93,\ldots \}&\operatorname {A109300} (5)\end{array}}$

Wiki Table

${\text{Table in which the}}~h^{\text{th}}~{\text{row lists the positive integers of rote height}}~h$
h	m such that rhig(m) = A109301(m) = h
0	1
1	2
2	3 4 6 9 12 18 36
3	5 7 8 10 13 14 15 16 20 21 23 24 25 26 27 28 30 35 37 39 40 42 45 46 48 49 50 52 54 56 60 61 63 64 65 69 70 72 74 75 78 80 81 84 90 91 92 98 100 …
4	11 17 19 22 29 32 33 34 38 41 43 44 47 51 53 55 57 58 66 68 71 73 76 77 82 83 85 86 87 88 89 94 95 96 97 99 …
5	31 59 62 67 79 93 …
Smallest m in the h^th row = A007097. Number of values in the h^th row = A109300(h). Number of values up through the h^th row = A050924(h + 1).

ASCII

 Example

    * Table in which the h^th row lists the positive integers of rote height h:
    * h | m such that rhig(m) = A109301(m) = h
    * --+------------------------------------------------------
    * 0 | 1
    * --+------------------------------------------------------
    * 1 | 2
    * --+------------------------------------------------------
    * 2 | 3 4 6 9 12 18 36
    * --+------------------------------------------------------
    * 3 | 5 7 8 10 13 14 15 16 20 21 23 24 25 26 27 28 30
    *   | 35 37 39 40 42 45 46 48 49 50 52 54 56 60 61 63
    *   | 64 65 69 70 72 74 75 78 80 81 84 90 91 92 98 100 ...
    * --+------------------------------------------------------
    * 4 | 11 17 19 22 29 32 33 34 38 41 43 44 47 51 53 55
    *   | 57 58 66 68 71 73 76 77 82 83 85 86 87 88 89 94
    *   | 95 96 97 99 ...
    * --+------------------------------------------------------
    * 5 | 31 59 62 67 79 93 ...
    * --+------------------------------------------------------
    * First column = A007097. Count in h^th row = A109300(h).
    * Cumulative count up through the h^th row = A050924(h+1).

A111792

Positive integers sorted by rote weight (A062537) and rote height (A109301).

TeX Array

${\begin{array}{rcl|rr|r|r}a&&\operatorname {code} &g&h&s&t\\\hline 1&=&\varnothing &0&0&1&1\\\hline 2&=&1\!:\!1&1&1&1&1\\\hline 3&=&2\!:\!1&2&2&&\\4&=&1\!:\!2&2&2&2&2\\\hline 6&=&1\!:\!1~~2\!:\!1&3&2&&\\9&=&2\!:\!2&3&2&2&\\\hline 5&=&3\!:\!1&3&3&&\\7&=&4\!:\!1&3&3&&\\8&=&1\!:\!3&3&3&&\\16&=&1\!:\!4&3&3&4&6\\\hline 12&=&1\!:\!2~~2\!:\!1&4&2&&\\18&=&1\!:\!1~~2\!:\!2&4&2&2&\\\hline 10&=&1\!:\!1~~3\!:\!1&4&3&&\\13&=&6\!:\!1&4&3&&\\14&=&1\!:\!1~~4\!:\!1&4&3&&\\23&=&9\!:\!1&4&3&&\\25&=&3\!:\!2&4&3&&\\27&=&2\!:\!3&4&3&&\\49&=&4\!:\!2&4&3&&\\64&=&1\!:\!6&4&3&&\\81&=&2\!:\!4&4&3&&\\512&=&1\!:\!9&4&3&10&\\\hline 11&=&5\!:\!1&4&4&&\\17&=&7\!:\!1&4&4&&\\19&=&8\!:\!1&4&4&&\\32&=&1\!:\!5&4&4&&\\53&=&16\!:\!1&4&4&&\\128&=&1\!:\!7&4&4&&\\256&=&1\!:\!8&4&4&&\\65536&=&1\!:\!16&4&4&8&20\\\hline \end{array}}$

ASCII

 Example

    * Table of Integers, Primal Codes, Sort Parameters and Subtotals
    *     a   code    | g h | s | t
    * ----------------+-----+---+---
    *     1 = { }     | 0 0 | 1 | 1
    * ----------------+-----+---+---
    *     2 = 1:1     | 1 1 | 1 | 1
    * ----------------+-----+---+---
    *     3 = 2:1     | 2 2 |   |
    *     4 = 1:2     | 2 2 | 2 | 2
    * ----------------+-----+---+---
    *     6 = 1:1 2:1 | 3 2 |   |
    *     9 = 2:2     | 3 2 | 2 |
    * ----------------+-----+---+---
    *     5 = 3:1     | 3 3 |   |
    *     7 = 4:1     | 3 3 |   |
    *     8 = 1:3     | 3 3 |   |
    *    16 = 1:4     | 3 3 | 4 | 6
    * ----------------+-----+---+---
    *    12 = 1:2 2:1 | 4 2 |   |
    *    18 = 1:1 2:2 | 4 2 | 2 |
    * ----------------+-----+---+---
    *    10 = 1:1 3:1 | 4 3 |   |
    *    13 = 6:1     | 4 3 |   |
    *    14 = 1:1 4:1 | 4 3 |   |
    *    23 = 9:1     | 4 3 |   |
    *    25 = 3:2     | 4 3 |   |
    *    27 = 2:3     | 4 3 |   |
    *    49 = 4:2     | 4 3 |   |
    *    64 = 1:6     | 4 3 |   |
    *    81 = 2:4     | 4 3 |   |
    *   512 = 1:9     | 4 3 |10 |
    * ----------------+-----+---+---
    *    11 = 5:1     | 4 4 |   |
    *    17 = 7:1     | 4 4 |   |
    *    19 = 8:1     | 4 4 |   |
    *    32 = 1:5     | 4 4 |   |
    *    53 = 16:1    | 4 4 |   |
    *   128 = 1:7     | 4 4 |   |
    *   256 = 1:8     | 4 4 |   |
    * 65536 = 1:16    | 4 4 | 8 |20
    * ----------------+-----+---+---
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A111793

Triangle T(g, h) = number of rotes of weight g and height h, both in gammas.

TeX Array

${\begin{array}{l|rrrrrr}g\backslash h&0&1&2&3&4&5\\\hline 0&1\\1&&1\\2&&&2\\3&&&2&4\\4&&&2&10&8\\5&&&1&24&32&16\end{array}}$

ASCII

 Example

    * Table T(g, h), omitting zeros, starts out as follows:
    * g\h| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 | 1
    *  1 |     1
    *  2 |         2
    *  3 |         2   4
    *  4 |         2  10   8
    *  5 |         1  24  32  16

A111794

Integers whose rote weight and rote height are equal, sorted by the equated value.

TeX Array

${\begin{array}{l|l|r}j&S_{j}~=~\{m~:~\operatorname {g} (m)~=~\operatorname {h} (m)~=~j\}&|S_{j}|\\\hline \hline 0&\{1\}&1\\\hline 1&\{2\}&1\\\hline 2&\{3,4\}&2\\\hline 3&\{5,7,8,16\}&4\\\hline 4&\{11,17,19,32,53,128,256,65536\}&8\\\hline 5&\{31,59,67,131,241,719,1619,2048,131072,524288,821641,\\&4294967296,9007199254740992,2^{128},2^{256},2^{65536}\}&16\end{array}}$

Wiki Table

${\text{Table whose}}~j^{\text{th}}~{\text{row lists the integers}}~m~{\text{with}}~g(m)=h(m)=j$
j	m such that g(m) = h(m) = j
0	1
1	2
2	3 4
3	5 7 8 16
4	11 17 19 32 53 128 256 65536
5	31 59 67 131 241 719 1619 2048 131072 524288 821641 4294967296 9007199254740992 2¹²⁸ 2²⁵⁶ 2⁶⁵⁵³⁶

ASCII

 Example

    * Triangle whose j^th row lists the integers m with g(m) = h(m) = j
    * j | m such that g(m) = h(m) = j
    * --+-------------------------------------------------------
    * 0 | 1
    * 1 | 2
    * 2 | 3 4
    * 3 | 5 7 8 16
    * 4 | 11 17 19 32 53 128 256 65536
    * 5 | 31 59 67 131 241 719 1619 2048 131072 524288 821641
    *   | 4294967296 9007199254740992 2^128 2^256 2^65536

A111795

Positive integers whose rote weight and rote height are equal.

JPEG

${\begin{array}{l}\varnothing \\1\end{array}}$

${\begin{array}{l}1\!:\!1\\2\end{array}}$

${\begin{array}{l}2\!:\!1\\3\end{array}}$

${\begin{array}{l}1\!:\!2\\4\end{array}}$

${\begin{array}{l}3\!:\!1\\5\end{array}}$

${\begin{array}{l}4\!:\!1\\7\end{array}}$

${\begin{array}{l}1\!:\!3\\8\end{array}}$

${\begin{array}{l}5\!:\!1\\11\end{array}}$

${\begin{array}{l}1\!:\!4\\16\end{array}}$

${\begin{array}{l}7\!:\!1\\17\end{array}}$

${\begin{array}{l}8\!:\!1\\19\end{array}}$

${\begin{array}{l}11\!:\!1\\31\end{array}}$

${\begin{array}{l}1\!:\!5\\32\end{array}}$

${\begin{array}{l}16\!:\!1\\53\end{array}}$

${\begin{array}{l}17\!:\!1\\59\end{array}}$

ASCII

 Example

    * Tables of Rotes and Primal Codes for a(1) to a(9)
    *                                                              
    *                                                 o-o          
    *                                                 |            
    *                           o-o     o-o     o-o   o-o       o-o
    *                           |       |       |     |         |  
    *             o-o     o-o   o-o   o-o       o-o   o-o     o-o  
    *             |       |     |     |         |     |       |    
    *       o-o   o-o   o-o     o-o   o-o     o-o     o-o   o-o    
    *       |     |     |       |     |       |       |     |      
    * O     O     O     O       O     O       O       O     O      
    *                                                              
    * { }   1:1   2:1   1:2     3:1   4:1     1:3     5:1   1:4    
    *                                                              
    * 1     2     3     4       5     7       8       11    16     
    *

A111796

Positive integers sorted by rote weight (A062537) and omega (A001221).

TeX Array

${\begin{array}{rcl|rr|r|r}a&&\operatorname {code} &g&w&s&t\\\hline 1&=&\varnothing &0&0&1&1\\\hline 2&=&1\!:\!1&1&1&1&1\\\hline 3&=&2\!:\!1&2&1&&\\4&=&1\!:\!2&2&1&2&2\\\hline 5&=&3\!:\!1&3&1&&\\7&=&4\!:\!1&3&1&&\\8&=&1\!:\!3&3&1&&\\9&=&2\!:\!2&3&1&&\\16&=&1\!:\!4&3&1&5&\\\hline 6&=&1\!:\!1~~2\!:\!1&3&2&1&6\\\hline 11&=&5\!:\!1&4&1&&\\13&=&6\!:\!1&4&1&&\\17&=&7\!:\!1&4&1&&\\19&=&8\!:\!1&4&1&&\\23&=&9\!:\!1&4&1&&\\25&=&3\!:\!2&4&1&&\\27&=&2\!:\!3&4&1&&\\32&=&1\!:\!5&4&1&&\\49&=&4\!:\!2&4&1&&\\53&=&16\!:\!1&4&1&&\\64&=&1\!:\!6&4&1&&\\81&=&2\!:\!4&4&1&&\\128&=&1\!:\!7&4&1&&\\256&=&1\!:\!8&4&1&&\\512&=&1\!:\!9&4&1&&\\65536&=&1\!:\!16&4&1&16&\\\hline 10&=&1\!:\!1~~3\!:\!1&4&2&&\\12&=&1\!:\!2~~2\!:\!1&4&2&&\\14&=&1\!:\!1~~4\!:\!1&4&2&&\\18&=&1\!:\!1~~2\!:\!2&4&2&4&20\\\hline \end{array}}$

ASCIII

 Example

    * Table of Integers, Primal Codes, Sort Parameters and Subtotals
    *     a   code    | g w | s | t
    * ----------------+-----+---+---
    *     1 = { }     | 0 0 | 1 | 1
    * ----------------+-----+---+---
    *     2 = 1:1     | 1 1 | 1 | 1
    * ----------------+-----+---+---
    *     3 = 2:1     | 2 1 |   |
    *     4 = 1:2     | 2 1 | 2 | 2
    * ----------------+-----+---+---
    *     5 = 3:1     | 3 1 |   |
    *     7 = 4:1     | 3 1 |   |
    *     8 = 1:3     | 3 1 |   |
    *     9 = 2:2     | 3 1 |   |
    *    16 = 1:4     | 3 1 | 5 |
    * ----------------+-----+---+---
    *     6 = 1:1 2:1 | 3 2 | 1 | 6
    * ----------------+-----+---+---
    *    11 = 5:1     | 4 1 |   |
    *    13 = 6:1     | 4 1 |   |
    *    17 = 7:1     | 4 1 |   |
    *    19 = 8:1     | 4 1 |   |
    *    23 = 9:1     | 4 1 |   |
    *    25 = 3:2     | 4 1 |   |
    *    27 = 2:3     | 4 1 |   |
    *    32 = 1:5     | 4 1 |   |
    *    49 = 4:2     | 4 1 |   |
    *    53 = 16:1    | 4 1 |   |
    *    64 = 1:6     | 4 1 |   |
    *    81 = 2:4     | 4 1 |   |
    *   128 = 1:7     | 4 1 |   |
    *   256 = 1:8     | 4 1 |   |
    *   512 = 1:9     | 4 1 |   |
    * 65536 = 1:16    | 4 1 |16 |
    * ----------------+-----+---+---
    *    10 = 1:1 3:1 | 4 2 |   |
    *    12 = 1:2 2:1 | 4 2 |   |
    *    14 = 1:1 4:1 | 4 2 |   |
    *    18 = 1:1 2:2 | 4 2 | 4 |20
    * ----------------+-----+---+---
    * a = this sequence
    * g = rote weight in gammas = A062537
    * w = rote wayage in gammas = A001221
    * s = count in (g, w) class = A111797
    * t = count in weight class = A061396

A111797

Triangle T(g, w) = number of rotes of weight g and wayage w.

TeX Array

${\begin{array}{l|rrrrrr}g\backslash w&0&1&2&3&4&5\\\hline 0&1\\1&&1\\2&&2\\3&&5&1\\4&&16&4\\5&&56&17\end{array}}$

ASCII

 Example

    * Table T(g, w), omitting zeros, begins as follows:
    * g\w| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 | 1
    *  1 |     1
    *  2 |     2
    *  3 |     5   1
    *  4 |    16   4
    *  5 |    56  17

A111798

Positive integers sorted by rote height (A109301) and omega (A001221).

TeX Array

${\begin{array}{l|r|rr|r|r}{\text{Primal Function}}&{\text{Primal Code}}~=~a&h&w&s&t\\\hline \varnothing &1&0&0&1&1\\\hline 1\!:\!1&2&1&1&1&1\\\hline 2\!:\!1&3&2&1&&\\1\!:\!2&4&2&1&&\\2\!:\!2&9&2&1&3&\\\hline 1\!:\!1~~2\!:\!1&6&2&2&&\\1\!:\!2~~2\!:\!1&12&2&2&&\\1\!:\!1~~2\!:\!2&18&2&2&&\\1\!:\!2~~2\!:\!2&36&2&2&4&7\\\hline &&&&&\\1\!:\!3&8&3&1&&\\1\!:\!4&16&3&1&&\\1\!:\!6&64&3&1&&\\1\!:\!9&512&3&1&&\\1\!:\!12&4096&3&1&&\\1\!:\!18&262144&3&1&&\\1\!:\!36&68719476736&3&1&&\\&&&&&\\2\!:\!3&27&3&1&&\\2\!:\!4&81&3&1&&\\2\!:\!6&729&3&1&&\\2\!:\!9&19683&3&1&&\\2\!:\!12&531441&3&1&&\\2\!:\!18&387420489&3&1&&\\2\!:\!36&150094635296999121&3&1&&\\&&&&&\\3\!:\!1&5&3&1&&\\4\!:\!1&7&3&1&&\\6\!:\!1&13&3&1&&\\9\!:\!1&23&3&1&&\\12\!:\!1&37&3&1&&\\18\!:\!1&61&3&1&&\\36\!:\!1&151&3&1&&\\&&&&&\\3\!:\!2&25&3&1&&\\4\!:\!2&49&3&1&&\\6\!:\!2&169&3&1&&\\9\!:\!2&529&3&1&&\\12\!:\!2&1369&3&1&&\\18\!:\!2&3721&3&1&&\\36\!:\!2&22801&3&1&&\\&&&&&\\3\!:\!3&125&3&1&&\\3\!:\!4&625&3&1&&\\3\!:\!6&15625&3&1&&\\3\!:\!9&1953125&3&1&&\\3\!:\!12&244140625&3&1&&\\3\!:\!18&3814697265625&3&1&&\\3\!:\!36&14551915228366851806640625&3&1&&\\&&&&&\\4\!:\!3&343&3&1&&\\4\!:\!4&2401&3&1&&\\4\!:\!6&117649&3&1&&\\4\!:\!9&40353607&3&1&&\\4\!:\!12&13841287201&3&1&&\\4\!:\!18&1628413597910449&3&1&&\\4\!:\!36&2651730845859653471779023381601&3&1&&\\&&&&&\\6\!:\!3&2197&3&1&&\\6\!:\!4&28561&3&1&&\\6\!:\!6&4826809&3&1&&\\6\!:\!9&10604499373&3&1&&\\6\!:\!12&23298085122481&3&1&&\\6\!:\!18&112455406951957393129&3&1&&\\6\!:\!36&13^{36}&3&1&&\\&&&&&\\9\!:\!3&12167&3&1&&\\9\!:\!4&279841&3&1&&\\9\!:\!6&148035889&3&1&&\\9\!:\!9&1801152661463&3&1&&\\9\!:\!12&21914624432020321&3&1&&\\9\!:\!18&3244150909895248285300369&3&1&&\\9\!:\!36&23^{36}&3&1&&\\&&&&&\\12\!:\!3&50653&3&1&&\\12\!:\!4&1874161&3&1&&\\12\!:\!6&2565726409&3&1&&\\12\!:\!9&129961739795077&3&1&&\\12\!:\!12&6582952005840035281&3&1&&\\12\!:\!18&16890053810563300749953435929&3&1&&\\12\!:\!36&37^{36}&3&1&&\\&&&&&\\18\!:\!3&226981&3&1&&\\18\!:\!4&13845841&3&1&&\\18\!:\!6&51520374361&3&1&&\\18\!:\!9&11694146092834141&3&1&&\\18\!:\!12&2654348974297586158321&3&1&&\\18\!:\!18&136753052840548005895349735207881&3&1&&\\18\!:\!36&61^{36}&3&1&&\\&&&&&\\36\!:\!3&3442951&3&1&&\\36\!:\!4&519885601&3&1&&\\36\!:\!6&11853911588401&3&1&&\\36\!:\!9&40812436757196811351&3&1&&\\36\!:\!12&140515219945627518837736801&3&1&&\\36\!:\!18&151^{18}&3&1&&\\36\!:\!36&151^{36}&3&1&77&\\\hline \end{array}}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * Primal Function |           Primal Code     =     a | h w | s | t
    * ----------------+-----------------------------------+-----+---+---
    * { }             |                                 1 | 0 0 | 1 | 1
    * ----------------+-----------------------------------+-----+---+---
    * 1:1             |                                 2 | 1 1 | 1 | 1
    * ----------------+-----------------------------------+-----+---+---
    * 2:1             |                                 3 | 2 1 |   |
    * 1:2             |                                 4 | 2 1 |   |
    * 2:2             |                                 9 | 2 1 | 3 |
    * ----------------+-----------------------------------+-----+---+---
    * 1:1 2:1         |                                 6 | 2 2 |   |
    * 1:2 2:1         |                                12 | 2 2 |   |
    * 1:1 2:2         |                                18 | 2 2 |   |
    * 1:2 2:2         |                                36 | 2 2 | 4 | 7
    * ----------------+-----------------------------------+-----+---+---
    *                 |                                   |     |   |
    * 1:3             |                                 8 | 3 1 |   |
    * 1:4             |                                16 | 3 1 |   |
    * 1:6             |                                64 | 3 1 |   |
    * 1:9             |                               512 | 3 1 |   |
    * 1:12            |                              4096 | 3 1 |   |
    * 1:18            |                            262144 | 3 1 |   |
    * 1:36            |                       68719476736 | 3 1 |   |
    *                 |                                   |     |   |
    * 2:3             |                                27 | 3 1 |   |
    * 2:4             |                                81 | 3 1 |   |
    * 2:6             |                               729 | 3 1 |   |
    * 2:9             |                             19683 | 3 1 |   |
    * 2:12            |                            531441 | 3 1 |   |
    * 2:18            |                         387420489 | 3 1 |   |
    * 2:36            |                150094635296999121 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:1             |                                 5 | 3 1 |   |
    * 4:1             |                                 7 | 3 1 |   |
    * 6:1             |                                13 | 3 1 |   |
    * 9:1             |                                23 | 3 1 |   |
    * 12:1            |                                37 | 3 1 |   |
    * 18:1            |                                61 | 3 1 |   |
    * 36:1            |                               151 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:2             |                                25 | 3 1 |   |
    * 4:2             |                                49 | 3 1 |   |
    * 6:2             |                               169 | 3 1 |   |
    * 9:2             |                               529 | 3 1 |   |
    * 12:2            |                              1369 | 3 1 |   |
    * 18:2            |                              3721 | 3 1 |   |
    * 36:2            |                             22801 | 3 1 |   |
    *                 |                                   |     |   |
    * 3:3             |                               125 | 3 1 |   |
    * 3:4             |                               625 | 3 1 |   |
    * 3:6             |                             15625 | 3 1 |   |
    * 3:9             |                           1953125 | 3 1 |   |
    * 3:12            |                         244140625 | 3 1 |   |
    * 3:18            |                     3814697265625 | 3 1 |   |
    * 3:36            |        14551915228366851806640625 | 3 1 |   |
    *                 |                                   |     |   |
    * 4:3             |                               343 | 3 1 |   |
    * 4:4             |                              2401 | 3 1 |   |
    * 4:6             |                            117649 | 3 1 |   |
    * 4:9             |                          40353607 | 3 1 |   |
    * 4:12            |                       13841287201 | 3 1 |   |
    * 4:18            |                  1628413597910449 | 3 1 |   |
    * 4:36            |   2651730845859653471779023381601 | 3 1 |   |
    *                 |                                   |     |   |
    * 6:3             |                              2197 | 3 1 |   |
    * 6:4             |                             28561 | 3 1 |   |
    * 6:6             |                           4826809 | 3 1 |   |
    * 6:9             |                       10604499373 | 3 1 |   |
    * 6:12            |                    23298085122481 | 3 1 |   |
    * 6:18            |             112455406951957393129 | 3 1 |   |
    * 6:36            |                             13^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 9:3             |                             12167 | 3 1 |   |
    * 9:4             |                            279841 | 3 1 |   |
    * 9:6             |                         148035889 | 3 1 |   |
    * 9:9             |                     1801152661463 | 3 1 |   |
    * 9:12            |                 21914624432020321 | 3 1 |   |
    * 9:18            |         3244150909895248285300369 | 3 1 |   |
    * 9:36            |                             23^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 12:3            |                             50653 | 3 1 |   |
    * 12:4            |                           1874161 | 3 1 |   |
    * 12:6            |                        2565726409 | 3 1 |   |
    * 12:9            |                   129961739795077 | 3 1 |   |
    * 12:12           |               6582952005840035281 | 3 1 |   |
    * 12:18           |     16890053810563300749953435929 | 3 1 |   |
    * 12:36           |                             37^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 18:3            |                            226981 | 3 1 |   |
    * 18:4            |                          13845841 | 3 1 |   |
    * 18:6            |                       51520374361 | 3 1 |   |
    * 18:9            |                 11694146092834141 | 3 1 |   |
    * 18:12           |            2654348974297586158321 | 3 1 |   |
    * 18:18           | 136753052840548005895349735207881 | 3 1 |   |
    * 18:36           |                             61^36 | 3 1 |   |
    *                 |                                   |     |   |
    * 36:3            |                           3442951 | 3 1 |   |
    * 36:4            |                         519885601 | 3 1 |   |
    * 36:6            |                    11853911588401 | 3 1 |   |
    * 36:9            |              40812436757196811351 | 3 1 |   |
    * 36:12           |       140515219945627518837736801 | 3 1 |   |
    * 36:18           |                            151^18 | 3 1 |   |
    * 36:36           |                            151^36 | 3 1 |77 |
    * ----------------+-----------------------------------+-----+---+---
    * The last part is left unsorted to show the method of construction.
    * a (when sorted ) = this sequence
    * h = rote height in gammas = A109301
    * w = rote wayage in gammas = A001221
    * s = count in (h, w) class = A111799
    * t = count in height class = A109300

A111799

Triangle T(h, w) = number of rotes of height h and wayage w.

TeX Array

${\begin{array}{l|*{10}{r}}h\backslash w&0&1&2&3&4&5&6&7&8&9\\\hline 0&1\\1&&1\\2&&3&4\\3&&77&?&?&?&?&?&?&?&?\end{array}}$

ASCII

 Example

    * Table T(h, w), omitting zeros, begins as follows:
    * h\w| 0   1   2   3   4   5   6   7   8   9
    * ---+---------------------------------------
    *  0 | 1
    *  1 |     1
    *  2 |     3   4
    *  3 |    77   ?   ?   ?   ?   ?   ?   ?   ?

A111800

Order of the rote (rooted odd tree with only exponent symmetries) for n.

TeX + JPEG

${\text{Writing}}~\operatorname {prime} (i)^{j}~{\text{as}}~i\!:\!j,2500=4\cdot 625=2^{2}5^{4}=1\!:\!2~~3\!:\!4~{\text{has the following rote:}}$

${\text{So}}~a(2500)=a(1\!:\!2~~3\!:\!4)=a(1)+a(2)+a(3)+a(4)+1=1+3+5+5+1=15.$

ASCII

 Example

    * Writing prime(i)^j as i:j and using equal signs between identified nodes:
    * 2500 = 4 * 625 = 2^2 5^4 = 1:2 3:4 has the following rote:
    *                
    *       o-o   o-o
    *       |     |  
    *   o-o o-o o-o  
    *   |   |   |    
    * o-o   o---o    
    * |     |        
    * O=====O        
    *                
    * So a(2500) = a(1:2 3:4) = a(1)+a(2)+a(3)+a(4)+1 = 1+3+5+5+1 = 15.

A111801

Numbers that have a positive primal code characteristic, that is, positive integers j for which A108352(j) > 0.

A112095

Positive integers sorted by rote weight, rote height, and rote wayage.

TeX Array

${\begin{array}{l|r|rrr|r|r|r}\hline \hline {\text{Primal Function}}&{\text{Primal Code}}~=~a&g&h&w&r&s&t\\\hline \hline \varnothing &1&0&0&0&1&1&1\\\hline \hline 1\!:\!1&2&1&1&1&1&1&1\\\hline \hline 2\!:\!1&3&2&2&1&&&\\1\!:\!2&4&2&2&1&2&2&2\\\hline \hline 2\!:\!2&9&3&2&1&1&&\\\hline 1\!:\!1~~2\!:\!1&6&3&2&2&1&2&\\\hline 3\!:\!1&5&3&3&1&&&\\4\!:\!1&7&3&3&1&&&\\1\!:\!3&8&3&3&1&&&\\1\!:\!4&16&3&3&1&4&4&6\\\hline \hline 1\!:\!2~~2\!:\!1&12&4&2&2&&&\\1\!:\!1~~2\!:\!2&18&4&2&2&2&2&\\\hline 6\!:\!1&13&4&3&1&&&\\9\!:\!1&23&4&3&1&&&\\3\!:\!2&25&4&3&1&&&\\2\!:\!3&27&4&3&1&&&\\4\!:\!2&49&4&3&1&&&\\1\!:\!6&64&4&3&1&&&\\2\!:\!4&81&4&3&1&&&\\1\!:\!9&512&4&3&1&8&&\\\hline 1\!:\!1~~3\!:\!1&10&4&3&2&&&\\1\!:\!1~~4\!:\!1&14&4&3&2&2&10&\\\hline 5\!:\!1&11&4&4&1&&&\\7\!:\!1&17&4&4&1&&&\\8\!:\!1&19&4&4&1&&&\\1\!:\!5&32&4&4&1&&&\\16\!:\!1&53&4&4&1&&&\\1\!:\!7&128&4&4&1&&&\\1\!:\!8&256&4&4&1&&&\\1\!:\!16&65536&4&4&1&8&8&20\\\hline \hline 1\!:\!2~~2\!:\!2&36&5&2&2&1&1&\\\hline 12\!:\!1&37&5&3&1&&&\\18\!:\!1&61&5&3&1&&&\\3\!:\!3&125&5&3&1&&&\\6\!:\!2&169&5&3&1&&&\\4\!:\!3&343&5&3&1&&&\\9\!:\!2&529&5&3&1&&&\\3\!:\!4&625&5&3&1&&&\\2\!:\!6&729&5&3&1&&&\\4\!:\!4&2401&5&3&1&&&\\1\!:\!12&4096&5&3&1&&&\\2\!:\!9&19683&5&3&1&&&\\1\!:\!18&262144&5&3&1&12&&\\\hline 2\!:\!1~~3\!:\!1&15&5&3&2&&&\\1\!:\!2~~3\!:\!1&20&5&3&2&&&\\2\!:\!1~~4\!:\!1&21&5&3&2&&&\\1\!:\!3~~2\!:\!1&24&5&3&2&&&\\1\!:\!1~~6\!:\!1&26&5&3&2&&&\\1\!:\!2~~4\!:\!1&28&5&3&2&&&\\1\!:\!1~~9\!:\!1&46&5&3&2&&&\\1\!:\!4~~2\!:\!1&48&5&3&2&&&\\1\!:\!1~~3\!:\!2&50&5&3&2&&&\\1\!:\!1~~2\!:\!3&54&5&3&2&&&\\1\!:\!1~~4\!:\!2&98&5&3&2&&&\\1\!:\!1~~2\!:\!4&162&5&3&2&12&24&\\\hline 10\!:\!1&29&5&4&1&&&\\13\!:\!1&41&5&4&1&&&\\14\!:\!1&43&5&4&1&&&\\23\!:\!1&83&5&4&1&&&\\25\!:\!1&97&5&4&1&&&\\27\!:\!1&103&5&4&1&&&\\5\!:\!2&121&5&4&1&&&\\49\!:\!1&227&5&4&1&&&\\2\!:\!5&243&5&4&1&&&\\7\!:\!2&289&5&4&1&&&\\64\!:\!1&311&5&4&1&&&\\8\!:\!2&361&5&4&1&&&\\81\!:\!1&419&5&4&1&&&\\1\!:\!10&1024&5&4&1&&&\\2\!:\!7&2187&5&4&1&&&\\16\!:\!2&2809&5&4&1&&&\\512\!:\!1&3671&5&4&1&&&\\2\!:\!8&6561&5&4&1&&&\\1\!:\!13&8192&5&4&1&&&\\1\!:\!14&16384&5&4&1&&&\\1\!:\!23&8388608&5&4&1&&&\\1\!:\!25&33554432&5&4&1&&&\\2\!:\!16&43046721&5&4&1&&&\\1\!:\!27&134217728&5&4&1&&&\\1\!:\!49&562949953421312&5&4&1&&&\\1\!:\!64&18446744073709551616&5&4&1&&&\\1\!:\!81&2417851639229258349412352&5&4&1&&&\\1\!:\!512&2^{512}&5&4&1&28&&\\\hline 1\!:\!1~~5\!:\!1&22&5&4&2&&&\\1\!:\!1~~7\!:\!1&34&5&4&2&&&\\1\!:\!1~~8\!:\!1&38&5&4&2&&&\\1\!:\!1~~16\!:\!1&106&5&4&2&4&32&\\\hline 11\!:\!1&31&5&5&1&&&\\17\!:\!1&59&5&5&1&&&\\19\!:\!1&67&5&5&1&&&\\32\!:\!1&131&5&5&1&&&\\53\!:\!1&241&5&5&1&&&\\128\!:\!1&719&5&5&1&&&\\256\!:\!1&1619&5&5&1&&&\\1\!:\!11&2048&5&5&1&&&\\1\!:\!17&131072&5&5&1&&&\\1\!:\!19&524288&5&5&1&&&\\65536\!:\!1&821641&5&5&1&&&\\1\!:\!32&4294967296&5&5&1&&&\\1\!:\!53&9007199254740992&5&5&1&&&\\1\!:\!128&2^{128}&5&5&1&&&\\1\!:\!256&2^{256}&5&5&1&&&\\1\!:\!65536&2^{65536}&5&5&1&16&16&73\\\hline \hline \end{array}}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g h w | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 1 |   |   |
    * 1:2             |                         4 | 2 2 1 | 2 | 2 | 2
    * ================================================================
    * 2:2             |                         9 | 3 2 1 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 2:1         |                         6 | 3 2 2 | 1 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 3 1 |   |   |
    * 4:1             |                         7 | 3 3 1 |   |   |
    * 1:3             |                         8 | 3 3 1 |   |   |
    * 1:4             |                        16 | 3 3 1 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 2 2 |   |   |
    * 1:1 2:2         |                        18 | 4 2 2 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 3 1 |   |   |
    * 9:1             |                        23 | 4 3 1 |   |   |
    * 3:2             |                        25 | 4 3 1 |   |   |
    * 2:3             |                        27 | 4 3 1 |   |   |
    * 4:2             |                        49 | 4 3 1 |   |   |
    * 1:6             |                        64 | 4 3 1 |   |   |
    * 2:4             |                        81 | 4 3 1 |   |   |
    * 1:9             |                       512 | 4 3 1 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 3 2 |   |   |
    * 1:1 4:1         |                        14 | 4 3 2 | 2 |10 |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 4 1 |   |   |
    * 7:1             |                        17 | 4 4 1 |   |   |
    * 8:1             |                        19 | 4 4 1 |   |   |
    * 1:5             |                        32 | 4 4 1 |   |   |
    * 16:1            |                        53 | 4 4 1 |   |   |
    * 1:7             |                       128 | 4 4 1 |   |   |
    * 1:8             |                       256 | 4 4 1 |   |   |
    * 1:16            |                     65536 | 4 4 1 | 8 | 8 |20
    * ================================================================
    * 1:2 2:2         |                        36 | 5 2 2 | 1 | 1 |
    * ----------------+---------------------------+-------+---+---+---
    * 12:1            |                        37 | 5 3 1 |   |   |
    * 18:1            |                        61 | 5 3 1 |   |   |
    * 3:3             |                       125 | 5 3 1 |   |   |
    * 6:2             |                       169 | 5 3 1 |   |   |
    * 4:3             |                       343 | 5 3 1 |   |   |
    * 9:2             |                       529 | 5 3 1 |   |   |
    * 3:4             |                       625 | 5 3 1 |   |   |
    * 2:6             |                       729 | 5 3 1 |   |   |
    * 4:4             |                      2401 | 5 3 1 |   |   |
    * 1:12            |                      4096 | 5 3 1 |   |   |
    * 2:9             |                     19683 | 5 3 1 |   |   |
    * 1:18            |                    262144 | 5 3 1 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 2:1 3:1         |                        15 | 5 3 2 |   |   |
    * 1:2 3:1         |                        20 | 5 3 2 |   |   |
    * 2:1 4:1         |                        21 | 5 3 2 |   |   |
    * 1:3 2:1         |                        24 | 5 3 2 |   |   |
    * 1:1 6:1         |                        26 | 5 3 2 |   |   |
    * 1:2 4:1         |                        28 | 5 3 2 |   |   |
    * 1:1 9:1         |                        46 | 5 3 2 |   |   |
    * 1:4 2:1         |                        48 | 5 3 2 |   |   |
    * 1:1 3:2         |                        50 | 5 3 2 |   |   |
    * 1:1 2:3         |                        54 | 5 3 2 |   |   |
    * 1:1 4:2         |                        98 | 5 3 2 |   |   |
    * 1:1 2:4         |                       162 | 5 3 2 |12 |24 |
    * ----------------+---------------------------+-------+---+---+---
    * 10:1            |                        29 | 5 4 1 |   |   |
    * 13:1            |                        41 | 5 4 1 |   |   |
    * 14:1            |                        43 | 5 4 1 |   |   |
    * 23:1            |                        83 | 5 4 1 |   |   |
    * 25:1            |                        97 | 5 4 1 |   |   |
    * 27:1            |                       103 | 5 4 1 |   |   |
    * 5:2             |                       121 | 5 4 1 |   |   |
    * 49:1            |                       227 | 5 4 1 |   |   |
    * 2:5             |                       243 | 5 4 1 |   |   |
    * 7:2             |                       289 | 5 4 1 |   |   |
    * 64:1            |                       311 | 5 4 1 |   |   |
    * 8:2             |                       361 | 5 4 1 |   |   |
    * 81:1            |                       419 | 5 4 1 |   |   |
    * 1:10            |                      1024 | 5 4 1 |   |   |
    * 2:7             |                      2187 | 5 4 1 |   |   |
    * 16:2            |                      2809 | 5 4 1 |   |   |
    * 512:1           |                      3671 | 5 4 1 |   |   |
    * 2:8             |                      6561 | 5 4 1 |   |   |
    * 1:13            |                      8192 | 5 4 1 |   |   |
    * 1:14            |                     16384 | 5 4 1 |   |   |
    * 1:23            |                   8388608 | 5 4 1 |   |   |
    * 1:25            |                  33554432 | 5 4 1 |   |   |
    * 2:16            |                  43046721 | 5 4 1 |   |   |
    * 1:27            |                 134217728 | 5 4 1 |   |   |
    * 1:49            |           562949953421312 | 5 4 1 |   |   |
    * 1:64            |      18446744073709551616 | 5 4 1 |   |   |
    * 1:81            | 2417851639229258349412352 | 5 4 1 |   |   |
    * 1:512           |                     2^512 | 5 4 1 |28 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 5:1         |                        22 | 5 4 2 |   |   |
    * 1:1 7:1         |                        34 | 5 4 2 |   |   |
    * 1:1 8:1         |                        38 | 5 4 2 |   |   |
    * 1:1 16:1        |                       106 | 5 4 2 | 4 |32 |
    * ----------------+---------------------------+-------+---+---+---
    * 11:1            |                        31 | 5 5 1 |   |   |
    * 17:1            |                        59 | 5 5 1 |   |   |
    * 19:1            |                        67 | 5 5 1 |   |   |
    * 32:1            |                       131 | 5 5 1 |   |   |
    * 53:1            |                       241 | 5 5 1 |   |   |
    * 128:1           |                       719 | 5 5 1 |   |   |
    * 256:1           |                      1619 | 5 5 1 |   |   |
    * 1:11            |                      2048 | 5 5 1 |   |   |
    * 1:17            |                    131072 | 5 5 1 |   |   |
    * 1:19            |                    524288 | 5 5 1 |   |   |
    * 65536:1         |                    821641 | 5 5 1 |   |   |
    * 1:32            |                4294967296 | 5 5 1 |   |   |
    * 1:53            |          9007199254740992 | 5 5 1 |   |   |
    * 1:128           |                     2^128 | 5 5 1 |   |   |
    * 1:256           |                     2^256 | 5 5 1 |   |   |
    * 1:65536         |                   2^65536 | 5 5 1 |16 |16 |73
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * w = rote wayage in gammas = A001221
    * r = number in (g,h,w) set = A112096
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A112096

Tetrahedron T(g, h, w) = number of rotes of weight g, height h, wayage w.

TeX Array

${\begin{array}{l|*{9}{r}}g\backslash (h,w)&(0,0)&(1,1)&(2,1)&(2,2)&(3,1)&(3,2)&(4,1)&(4,2)&(5,1)\\\hline 0&1~\\1&&1~\\2&&&2~\\3&&&1~&1~&4~\\4&&&&2~&8~&2~&8~\\5&&&&1~&12~&12~&28~&4~&16~\end{array}}$

ASCII

 Example

    * Table T(g, h, w), omitting empty cells, starts out as follows:
    * g\(h,w) | (0,0) (1,1) (2,1) (2,2) (3,1) (3,2) (4,1) (4,2) (5,1)
    * --------+-------------------------------------------------------
    * 0       |   1
    * 1       |         1
    * 2       |               2
    * 3       |               1     1     4
    * 4       |                     2     8     2     8
    * 5       |                     1    12    12    28     4    16

A112480

Positive integers sorted by rote weight, rote wagage, and rote height.

TeX Array

${\begin{array}{l|r|rrr|r|r|r}\hline \hline {\text{Primal Function}}&{\text{Primal Code}}~=~a&g&w&h&r&s&t\\\hline \hline \varnothing &1&0&0&0&1&1&1\\\hline \hline 1\!:\!1&2&1&1&1&1&1&1\\\hline \hline 2\!:\!1&3&2&1&2&&&\\1\!:\!2&4&2&1&2&2&2&2\\\hline \hline 2\!:\!2&9&3&1&2&1&&\\\hline 3\!:\!1&5&3&1&3&&&\\4\!:\!1&7&3&1&3&&&\\1\!:\!3&8&3&1&3&&&\\1\!:\!4&16&3&1&3&4&5&\\\hline 1\!:\!1~~2\!:\!1&6&3&2&2&1&1&6\\\hline \hline 6\!:\!1&13&4&1&3&&&\\9\!:\!1&23&4&1&3&&&\\3\!:\!2&25&4&1&3&&&\\2\!:\!3&27&4&1&3&&&\\4\!:\!2&49&4&1&3&&&\\1\!:\!6&64&4&1&3&&&\\2\!:\!4&81&4&1&3&&&\\1\!:\!9&512&4&1&3&8&&\\\hline 5\!:\!1&11&4&1&4&&&\\7\!:\!1&17&4&1&4&&&\\8\!:\!1&19&4&1&4&&&\\1\!:\!5&32&4&1&4&&&\\16\!:\!1&53&4&1&4&&&\\1\!:\!7&128&4&1&4&&&\\1\!:\!8&256&4&1&4&&&\\1\!:\!16&65536&4&1&4&8&16&\\\hline 1\!:\!22\!:\!1&12&4&2&2&&&\\1\!:\!12\!:\!2&18&4&2&2&2&&\\\hline 1\!:\!13\!:\!1&10&4&2&3&&&\\1\!:\!14\!:\!1&14&4&2&3&2&4&20\\\hline \hline 12\!:\!1&37&5&1&3&&&\\18\!:\!1&61&5&1&3&&&\\3\!:\!3&125&5&1&3&&&\\6\!:\!2&169&5&1&3&&&\\4\!:\!3&343&5&1&3&&&\\9\!:\!2&529&5&1&3&&&\\3\!:\!4&625&5&1&3&&&\\2\!:\!6&729&5&1&3&&&\\4\!:\!4&2401&5&1&3&&&\\1\!:\!12&4096&5&1&3&&&\\2\!:\!9&19683&5&1&3&&&\\1\!:\!18&262144&5&1&3&12&&\\\hline 10\!:\!1&29&5&1&4&&&\\13\!:\!1&41&5&1&4&&&\\14\!:\!1&43&5&1&4&&&\\23\!:\!1&83&5&1&4&&&\\25\!:\!1&97&5&1&4&&&\\27\!:\!1&103&5&1&4&&&\\5\!:\!2&121&5&1&4&&&\\49\!:\!1&227&5&1&4&&&\\2\!:\!5&243&5&1&4&&&\\7\!:\!2&289&5&1&4&&&\\64\!:\!1&311&5&1&4&&&\\8\!:\!2&361&5&1&4&&&\\81\!:\!1&419&5&1&4&&&\\1\!:\!10&1024&5&1&4&&&\\2\!:\!7&2187&5&1&4&&&\\16\!:\!2&2809&5&1&4&&&\\512\!:\!1&3671&5&1&4&&&\\2\!:\!8&6561&5&1&4&&&\\1\!:\!13&8192&5&1&4&&&\\1\!:\!14&16384&5&1&4&&&\\1\!:\!23&8388608&5&1&4&&&\\1\!:\!25&33554432&5&1&4&&&\\2\!:\!16&43046721&5&1&4&&&\\1\!:\!27&134217728&5&1&4&&&\\1\!:\!49&562949953421312&5&1&4&&&\\1\!:\!64&18446744073709551616&5&1&4&&&\\1\!:\!81&2417851639229258349412352&5&1&4&&&\\1\!:\!512&2^{512}&5&1&4&28&&\\\hline 11\!:\!1&31&5&1&5&&&\\17\!:\!1&59&5&1&5&&&\\19\!:\!1&67&5&1&5&&&\\32\!:\!1&131&5&1&5&&&\\53\!:\!1&241&5&1&5&&&\\128\!:\!1&719&5&1&5&&&\\256\!:\!1&1619&5&1&5&&&\\1\!:\!11&2048&5&1&5&&&\\1\!:\!17&131072&5&1&5&&&\\1\!:\!19&524288&5&1&5&&&\\65536\!:\!1&821641&5&1&5&&&\\1\!:\!32&4294967296&5&1&5&&&\\1\!:\!53&9007199254740992&5&1&5&&&\\1\!:\!128&2^{128}&5&1&5&&&\\1\!:\!256&2^{256}&5&1&5&&&\\1\!:\!65536&2^{65536}&5&1&5&16&56&\\\hline 1\!:\!2~~2\!:\!2&36&5&2&2&1&&\\\hline 2\!:\!1~~3\!:\!1&15&5&2&3&&&\\1\!:\!2~~3\!:\!1&20&5&2&3&&&\\2\!:\!1~~4\!:\!1&21&5&2&3&&&\\1\!:\!3~~2\!:\!1&24&5&2&3&&&\\1\!:\!1~~6\!:\!1&26&5&2&3&&&\\1\!:\!2~~4\!:\!1&28&5&2&3&&&\\1\!:\!1~~9\!:\!1&46&5&2&3&&&\\1\!:\!4~~2\!:\!1&48&5&2&3&&&\\1\!:\!1~~3\!:\!2&50&5&2&3&&&\\1\!:\!1~~2\!:\!3&54&5&2&3&&&\\1\!:\!1~~4\!:\!2&98&5&2&3&&&\\1\!:\!1~~2\!:\!4&162&5&2&3&12&&\\\hline 1\!:\!1~~5\!:\!1&22&5&2&4&&&\\1\!:\!1~~7\!:\!1&34&5&2&4&&&\\1\!:\!1~~8\!:\!1&38&5&2&4&&&\\1\!:\!1~~16\!:\!1&106&5&2&4&4&17&73\\\hline \hline \end{array}}$

ASCII

 Example

    * Table of Primal Functions, Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g w h | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 1 2 |   |   |
    * 1:2             |                         4 | 2 1 2 | 2 | 2 | 2
    * ================================================================
    * 2:2             |                         9 | 3 1 2 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 1 3 |   |   |
    * 4:1             |                         7 | 3 1 3 |   |   |
    * 1:3             |                         8 | 3 1 3 |   |   |
    * 1:4             |                        16 | 3 1 3 | 4 | 5 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 2:1         |                         6 | 3 2 2 | 1 | 1 | 6
    * ================================================================
    * 6:1             |                        13 | 4 1 3 |   |   |
    * 9:1             |                        23 | 4 1 3 |   |   |
    * 3:2             |                        25 | 4 1 3 |   |   |
    * 2:3             |                        27 | 4 1 3 |   |   |
    * 4:2             |                        49 | 4 1 3 |   |   |
    * 1:6             |                        64 | 4 1 3 |   |   |
    * 2:4             |                        81 | 4 1 3 |   |   |
    * 1:9             |                       512 | 4 1 3 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 1 4 |   |   |
    * 7:1             |                        17 | 4 1 4 |   |   |
    * 8:1             |                        19 | 4 1 4 |   |   |
    * 1:5             |                        32 | 4 1 4 |   |   |
    * 16:1            |                        53 | 4 1 4 |   |   |
    * 1:7             |                       128 | 4 1 4 |   |   |
    * 1:8             |                       256 | 4 1 4 |   |   |
    * 1:16            |                     65536 | 4 1 4 | 8 |16 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:2 2:1         |                        12 | 4 2 2 |   |   |
    * 1:1 2:2         |                        18 | 4 2 2 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 2 3 |   |   |
    * 1:1 4:1         |                        14 | 4 2 3 | 2 | 4 |20
    * ================================================================
    * 12:1            |                        37 | 5 1 3 |   |   |
    * 18:1            |                        61 | 5 1 3 |   |   |
    * 3:3             |                       125 | 5 1 3 |   |   |
    * 6:2             |                       169 | 5 1 3 |   |   |
    * 4:3             |                       343 | 5 1 3 |   |   |
    * 9:2             |                       529 | 5 1 3 |   |   |
    * 3:4             |                       625 | 5 1 3 |   |   |
    * 2:6             |                       729 | 5 1 3 |   |   |
    * 4:4             |                      2401 | 5 1 3 |   |   |
    * 1:12            |                      4096 | 5 1 3 |   |   |
    * 2:9             |                     19683 | 5 1 3 |   |   |
    * 1:18            |                    262144 | 5 1 3 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 10:1            |                        29 | 5 1 4 |   |   |
    * 13:1            |                        41 | 5 1 4 |   |   |
    * 14:1            |                        43 | 5 1 4 |   |   |
    * 23:1            |                        83 | 5 1 4 |   |   |
    * 25:1            |                        97 | 5 1 4 |   |   |
    * 27:1            |                       103 | 5 1 4 |   |   |
    * 5:2             |                       121 | 5 1 4 |   |   |
    * 49:1            |                       227 | 5 1 4 |   |   |
    * 2:5             |                       243 | 5 1 4 |   |   |
    * 7:2             |                       289 | 5 1 4 |   |   |
    * 64:1            |                       311 | 5 1 4 |   |   |
    * 8:2             |                       361 | 5 1 4 |   |   |
    * 81:1            |                       419 | 5 1 4 |   |   |
    * 1:10            |                      1024 | 5 1 4 |   |   |
    * 2:7             |                      2187 | 5 1 4 |   |   |
    * 16:2            |                      2809 | 5 1 4 |   |   |
    * 512:1           |                      3671 | 5 1 4 |   |   |
    * 2:8             |                      6561 | 5 1 4 |   |   |
    * 1:13            |                      8192 | 5 1 4 |   |   |
    * 1:14            |                     16384 | 5 1 4 |   |   |
    * 1:23            |                   8388608 | 5 1 4 |   |   |
    * 1:25            |                  33554432 | 5 1 4 |   |   |
    * 2:16            |                  43046721 | 5 1 4 |   |   |
    * 1:27            |                 134217728 | 5 1 4 |   |   |
    * 1:49            |           562949953421312 | 5 1 4 |   |   |
    * 1:64            |      18446744073709551616 | 5 1 4 |   |   |
    * 1:81            | 2417851639229258349412352 | 5 1 4 |   |   |
    * 1:512           |                     2^512 | 5 1 4 |28 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 11:1            |                        31 | 5 1 5 |   |   |
    * 17:1            |                        59 | 5 1 5 |   |   |
    * 19:1            |                        67 | 5 1 5 |   |   |
    * 32:1            |                       131 | 5 1 5 |   |   |
    * 53:1            |                       241 | 5 1 5 |   |   |
    * 128:1           |                       719 | 5 1 5 |   |   |
    * 256:1           |                      1619 | 5 1 5 |   |   |
    * 1:11            |                      2048 | 5 1 5 |   |   |
    * 1:17            |                    131072 | 5 1 5 |   |   |
    * 1:19            |                    524288 | 5 1 5 |   |   |
    * 65536:1         |                    821641 | 5 1 5 |   |   |
    * 1:32            |                4294967296 | 5 1 5 |   |   |
    * 1:53            |          9007199254740992 | 5 1 5 |   |   |
    * 1:128           |                     2^128 | 5 1 5 |   |   |
    * 1:256           |                     2^256 | 5 1 5 |   |   |
    * 1:65536         |                   2^65536 | 5 1 5 |16 |56 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:2 2:2         |                        36 | 5 2 2 | 1 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 2:1 3:1         |                        15 | 5 2 3 |   |   |
    * 1:2 3:1         |                        20 | 5 2 3 |   |   |
    * 2:1 4:1         |                        21 | 5 2 3 |   |   |
    * 1:3 2:1         |                        24 | 5 2 3 |   |   |
    * 1:1 6:1         |                        26 | 5 2 3 |   |   |
    * 1:2 4:1         |                        28 | 5 2 3 |   |   |
    * 1:1 9:1         |                        46 | 5 2 3 |   |   |
    * 1:4 2:1         |                        48 | 5 2 3 |   |   |
    * 1:1 3:2         |                        50 | 5 2 3 |   |   |
    * 1:1 2:3         |                        54 | 5 2 3 |   |   |
    * 1:1 4:2         |                        98 | 5 2 3 |   |   |
    * 1:1 2:4         |                       162 | 5 2 3 |12 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 5:1         |                        22 | 5 2 4 |   |   |
    * 1:1 7:1         |                        34 | 5 2 4 |   |   |
    * 1:1 8:1         |                        38 | 5 2 4 |   |   |
    * 1:1 16:1        |                       106 | 5 2 4 | 4 |17 |73
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * w = rote wayage in gammas = A001221
    * h = rote height in gammas = A109301
    * r = number in (g,h,w) set = A112481
    * s = count in (g, w) class = A111797
    * t = count in weight class = A061396

A112481

Tetrahedron T(g, w, h) = number of rotes of weight g, wayage w, height h.

TeX Array

${\begin{array}{l|*{9}{r}}g\backslash (w,h)&(0,0)&(1,1)&(1,2)&&(1,3)&&(1,4)&&(1,5)\\&&&&(2,2)&&(2,3)&&(2,4)&\\\hline \hline 0&1~\\\hline 1&&1~\\\hline 2&&&2~\\\hline 3&&&1~&&4~\\&&&&1~&\\\hline 4&&&&&8~&&8~\\&&&&2~&&2~&\\\hline 5&&&&&12~&&28~&&16~\\&&&&1~&&12~&&4~&\\\hline \end{array}}$

ASCII

 Example

    * Table T(g, w, h), omitting empty cells, starts out as follows:
    * --------+-------------------------------------------------------
    * g\(w,h) | (0,0) (1,1) (1,2)       (1,3)       (1,4)       (1,5)
    *         |                   (2,2)       (2,3)       (2,4)      
    * ========+=======================================================
    * 0       |   1                                                  
    * --------+-------------------------------------------------------
    * 1       |         1                                            
    * --------+-------------------------------------------------------
    * 2       |               2                                      
    * --------+-------------------------------------------------------
    * 3       |               1           4                          
    * 3       |                     1                                
    * --------+-------------------------------------------------------
    * 4       |                           8           8              
    * 4       |                     2           2                    
    * --------+-------------------------------------------------------
    * 5       |                          12          28          16  
    * 5       |                     1          12           4        
    * --------+-------------------------------------------------------
    * Row sums = A111797. Horizontal section sums = A061396.

A112846

Number of riffs on n or fewer nodes. Number of rotes on 2n+1 or fewer nodes.

A112868

Positive integers sorted by rote weight and primal code characteristic.

TeX Array

${\begin{array}{l|r|rr|r|r}\hline \hline {\text{Primal Function}}&{\text{Primal Code}}~=~a&g&q&s&t\\\hline \hline \varnothing &1&0&1&1&1\\\hline \hline 1\!:\!1&2&1&0&1&1\\\hline \hline 2\!:\!1&3&2&2&&\\1\!:\!2&4&2&2&2&2\\\hline \hline 1\!:\!1~~2\!:\!1&6&3&0&&\\2\!:\!2&9&3&0&2&\\\hline 3\!:\!1&5&3&2&&\\4\!:\!1&7&3&2&&\\1\!:\!3&8&3&2&&\\1\!:\!4&16&3&2&4&6\\\hline \hline 1\!:\!1~~3\!:\!1&10&4&0&&\\1\!:\!2~~2\!:\!1&12&4&0&&\\1\!:\!1~~4\!:\!1&14&4&0&&\\1\!:\!1~~2\!:\!2&18&4&0&4&\\\hline 5\!:\!1&11&4&2&&\\6\!:\!1&13&4&2&&\\7\!:\!1&17&4&2&&\\8\!:\!1&19&4&2&&\\9\!:\!1&23&4&2&&\\3\!:\!2&25&4&2&&\\2\!:\!3&27&4&2&&\\1\!:\!5&32&4&2&&\\4\!:\!2&49&4&2&&\\16\!:\!1&53&4&2&&\\1\!:\!6&64&4&2&&\\2\!:\!4&81&4&2&&\\1\!:\!7&128&4&2&&\\1\!:\!8&256&4&2&&\\1\!:\!9&512&4&2&&\\1\!:\!16&65536&4&2&16&20\\\hline \hline \end{array}}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters, Subtotals
    * ==========================================================
    * Primal Function |       Primal Code   =   a | g q | s | t
    * ==========================================================
    * { }             |                         1 | 0 1 | 1 | 1
    * ==========================================================
    * 1:1             |                         2 | 1 0 | 1 | 1
    * ==========================================================
    * 2:1             |                         3 | 2 2 |   |
    * 1:2             |                         4 | 2 2 | 2 | 2
    * ==========================================================
    * 1:1 2:1         |                         6 | 3 0 |   |
    * 2:2             |                         9 | 3 0 | 2 |
    * ----------------+---------------------------+-----+---+---
    * 3:1             |                         5 | 3 2 |   |
    * 4:1             |                         7 | 3 2 |   |
    * 1:3             |                         8 | 3 2 |   |
    * 1:4             |                        16 | 3 2 | 4 | 6
    * ==========================================================
    * 1:1 3:1         |                        10 | 4 0 |   |
    * 1:2 2:1         |                        12 | 4 0 |   |
    * 1:1 4:1         |                        14 | 4 0 |   |
    * 1:1 2:2         |                        18 | 4 0 | 4 |
    * ----------------+---------------------------+-----+---+---
    * 5:1             |                        11 | 4 2 |   |
    * 6:1             |                        13 | 4 2 |   |
    * 7:1             |                        17 | 4 2 |   |
    * 8:1             |                        19 | 4 2 |   |
    * 9:1             |                        23 | 4 2 |   |
    * 3:2             |                        25 | 4 2 |   |
    * 2:3             |                        27 | 4 2 |   |
    * 1:5             |                        32 | 4 2 |   |
    * 4:2             |                        49 | 4 2 |   |
    * 16:1            |                        53 | 4 2 |   |
    * 1:6             |                        64 | 4 2 |   |
    * 2:4             |                        81 | 4 2 |   |
    * 1:7             |                       128 | 4 2 |   |
    * 1:8             |                       256 | 4 2 |   |
    * 1:9             |                       512 | 4 2 |   |
    * 1:16            |                     65536 | 4 2 |16 |20
    * ==========================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * q = primal code character = A108352
    * s = count in (g, q) class = A112869
    * t = count in weight class = A061396

A112869

Triangle T(g, q) = number of rotes of weight g and primal code characteristic q.

TeX Array

${\begin{array}{l|rrrrrr}g\backslash q&0&1&2&3&4&5\\\hline 0&&1\\1&1&\\2&&&2\\3&2&&4\\4&4&&16\\5&13&&56&4\end{array}}$

ASCII

 Example

    * Table T(g, q), omitting empty cells, begins as follows:
    * g\q| 0   1   2   3   4   5
    * ---+-----------------------
    *  0 |     1
    *  1 | 1
    *  2 |         2
    *  3 | 2       4
    *  4 | 4      16
    *  5 |13      56   4

A112870

Positive integers sorted by rote height and primal code characteristic.

TeX Array

${\begin{array}{l|r|rr|r|r}{\text{Primal Function}}&{\text{Primal Code}}~=~a&h&q&s&t\\\hline \varnothing &1&0&1&1&1\\\hline 1\!:\!1&2&1&0&1&1\\\hline 1\!:\!1~~2\!:\!1&6&2&0&&\\2\!:\!2&9&2&0&&\\1\!:\!2~~2\!:\!1&12&2&0&&\\1\!:\!1~~2\!:\!2&18&2&0&&\\1\!:\!2~~2\!:\!2&36&2&0&5&\\\hline 2\!:\!1&3&2&2&&\\1\!:\!2&4&2&2&2&7\\\hline \end{array}}$

ASCII

 Example

    * Primal Function | Primal Code = a | h q | s | t
    * ----------------+-----------------+-----+---+---
    * { }             |               1 | 0 1 | 1 | 1
    * ----------------+-----------------+-----+---+---
    * 1:1             |               2 | 1 0 | 1 | 1
    * ----------------+-----------------+-----+---+---
    * 1:1 2:1         |               6 | 2 0 |   |
    * 2:2             |               9 | 2 0 |   |
    * 1:2 2:1         |              12 | 2 0 |   |
    * 1:1 2:2         |              18 | 2 0 |   |
    * 1:2 2:2         |              36 | 2 0 | 5 |
    * ----------------+-----------------+-----+---+---
    * 2:1             |               3 | 2 2 |   |
    * 1:2             |               4 | 2 2 | 2 | 7
    * ----------------+-----------------+-----+---+---
    * a = this sequence
    * h = rote height in gammas = A109301
    * q = primal code character = A108352
    * s = count in (h, q) class = A112871
    * t = count in height class = A109300

A112871

Triangle T(h, q) = number of rotes of height h and quench q.

TeX Array

${\begin{array}{l|rrr}h\backslash q&0&1&2\\\hline 0&&1&\\1&1&&\\2&5&&2\\\end{array}}$

ASCII

 Example

    * Table T(h, q), omitting empty cells, begins as follows:
    * h\q| 0   1   2
    * ---+----------
    *  0 |     1    
    *  1 | 1        
    *  2 | 5       2
    * Row sums = A109300.

A112872

First differences of A061396.

A113197

Positive integers sorted by rote weight, rote height, and rote quench.

TeX Array

${\begin{array}{l|r|rrr|r|r|r}\hline \hline {\text{Primal Function}}&{\text{Primal Code}}~=~a&g&h&q&r&s&t\\\hline \hline \varnothing &1&0&0&1&1&1&1\\\hline \hline 1\!:\!1&2&1&1&0&1&1&1\\\hline \hline 2\!:\!1&3&2&2&2&&&\\1\!:\!2&4&2&2&2&2&2&2\\\hline \hline 1\!:\!1~~2\!:\!1&6&3&2&0&&&\\2\!:\!2&9&3&2&0&2&2&\\\hline 3\!:\!1&5&3&3&2&&&\\4\!:\!1&7&3&3&2&&&\\1\!:\!3&8&3&3&2&&&\\1\!:\!4&16&3&3&2&4&4&6\\\hline \hline 1\!:\!2~~2\!:\!1&12&4&2&0&&&\\1\!:\!1~~2\!:\!2&18&4&2&0&2&2&\\\hline 1\!:\!1~~3\!:\!1&10&4&3&0&&&\\1\!:\!1~~4\!:\!1&14&4&3&0&2&&\\\hline 6\!:\!1&13&4&3&2&&&\\9\!:\!1&23&4&3&2&&&\\3\!:\!2&25&4&3&2&&&\\2\!:\!3&27&4&3&2&&&\\4\!:\!2&49&4&3&2&&&\\1\!:\!6&64&4&3&2&&&\\2\!:\!4&81&4&3&2&&&\\1\!:\!9&512&4&3&2&8&10&\\\hline 5\!:\!1&11&4&4&2&&&\\7\!:\!1&17&4&4&2&&&\\8\!:\!1&19&4&4&2&&&\\1\!:\!5&32&4&4&2&&&\\16\!:\!1&53&4&4&2&&&\\1\!:\!7&128&4&4&2&&&\\1\!:\!8&256&4&4&2&&&\\1\!:\!16&65536&4&4&2&8&8&20\\\hline \hline \end{array}}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g h q | r | s | t
    * ================================================================
    * { }             |                         1 | 0 0 1 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 1 0 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 2 |   |   |
    * 1:2             |                         4 | 2 2 2 | 2 | 2 | 2
    * ================================================================
    * 1:1 2:1         |                         6 | 3 2 0 |   |   |
    * 2:2             |                         9 | 3 2 0 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 3 2 |   |   |
    * 4:1             |                         7 | 3 3 2 |   |   |
    * 1:3             |                         8 | 3 3 2 |   |   |
    * 1:4             |                        16 | 3 3 2 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 2 0 |   |   |
    * 1:1 2:2         |                        18 | 4 2 0 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 3 0 |   |   |
    * 1:1 4:1         |                        14 | 4 3 0 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 3 2 |   |   |
    * 9:1             |                        23 | 4 3 2 |   |   |
    * 3:2             |                        25 | 4 3 2 |   |   |
    * 2:3             |                        27 | 4 3 2 |   |   |
    * 4:2             |                        49 | 4 3 2 |   |   |
    * 1:6             |                        64 | 4 3 2 |   |   |
    * 2:4             |                        81 | 4 3 2 |   |   |
    * 1:9             |                       512 | 4 3 2 | 8 |10 |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 4 2 |   |   |
    * 7:1             |                        17 | 4 4 2 |   |   |
    * 8:1             |                        19 | 4 4 2 |   |   |
    * 1:5             |                        32 | 4 4 2 |   |   |
    * 16:1            |                        53 | 4 4 2 |   |   |
    * 1:7             |                       128 | 4 4 2 |   |   |
    * 1:8             |                       256 | 4 4 2 |   |   |
    * 1:16            |                     65536 | 4 4 2 | 8 | 8 |20
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * h = rote height in gammas = A109301
    * q = primal code character = A108352
    * r = number in (g,h,q) set = A113198
    * s = count in (g, h) class = A111793
    * t = count in weight class = A061396

A113198

Tetrahedron T(g, h, q) = number of rotes of weight g, height h, quench q.

TeX Array

${\begin{array}{l|*{10}{r}}\hline \\g\backslash (h,q)&(0,1)\\&&(0,1)\\&&&(2,0)&(2,2)\\&&&&&(3,0)&(3,2)&(3,3)\\&&&&&&&&(4,0)&(4,2)\\&&&&&&&&&&(5,2)\\\hline \hline 0&1~\\\hline 1&&1~\\\hline 2&&&&2~\\\hline 3&&&2~&&&\\&&&&&&4~\\\hline 4&&&2~&&&&&&\\&&&&&2~&8~&&&\\&&&&&&&&&8~\\\hline 5&&&1~&&&&&&&\\&&&&&8~&12~&4~&&&\\&&&&&&&&4~&28~&\\&&&&&&&&&&16~\\\hline \end{array}}$

ASCII

 Example

    * Table T(g, h, q), omitting empty cells, starts out as follows:
    * --------+------------------------------------------------------------
    * g\(h,q) | (0,1)                                                      
    *         |       (1,0)                                                
    *         |             (2,0) (2,2)                                    
    *         |                         (3,0) (3,2) (3,3)                  
    *         |                                           (4,0) (4,2)      
    *         |                                                       (5,2)
    * ========+============================================================
    * 0       |   1                                                        
    * --------+------------------------------------------------------------
    * 1       |         1                                                  
    * --------+------------------------------------------------------------
    * 2       |                     2                                      
    * --------+------------------------------------------------------------
    * 3       |               2                                            
    * 3       |                                 4                          
    * --------+------------------------------------------------------------
    * 4       |               2                                            
    * 4       |                           2     8                          
    * 4       |                                                   8        
    * --------+------------------------------------------------------------
    * 5       |               1                                            
    * 5       |                           8    12     4                    
    * 5       |                                             4    28        
    * 5       |                                                        16  
    * --------+------------------------------------------------------------
    * Row sums = A111793. Horizontal section sums = A061396.

A113199

Positive integers sorted by rote weight, rote quench, and rote height.

TeX Array

${\begin{array}{l|r|rrr|r|r|r}\hline \hline {\text{Primal Function}}&{\text{Primal Code}}~=~a&g&q&h&r&s&t\\\hline \hline \varnothing &1&0&1&0&1&1&1\\\hline \hline 1\!:\!1&2&1&0&1&1&1&1\\\hline \hline 2\!:\!1&3&2&2&2&&&\\1\!:\!2&4&2&2&2&2&2&2\\\hline \hline 1\!:\!1~~2\!:\!1&6&3&0&2&&&\\2\!:\!2&9&3&0&2&2&2&\\\hline 3\!:\!1&5&3&2&3&&&\\4\!:\!1&7&3&2&3&&&\\1\!:\!3&8&3&2&3&&&\\1\!:\!4&16&3&2&3&4&4&6\\\hline \hline 1\!:\!2~~2\!:\!1&12&4&0&2&&&\\1\!:\!1~~2\!:\!2&18&4&0&2&2&&\\\hline 1\!:\!1~~3\!:\!1&10&4&0&3&&&\\1\!:\!1~~4\!:\!1&14&4&0&3&2&4&\\\hline 6\!:\!1&13&4&2&3&&&\\9\!:\!1&23&4&2&3&&&\\3\!:\!2&25&4&2&3&&&\\2\!:\!3&27&4&2&3&&&\\4\!:\!2&49&4&2&3&&&\\1\!:\!6&64&4&2&3&&&\\2\!:\!4&81&4&2&3&&&\\1\!:\!9&512&4&2&3&8&&\\\hline 5\!:\!1&11&4&2&4&&&\\7\!:\!1&17&4&2&4&&&\\8\!:\!1&19&4&2&4&&&\\1\!:\!5&32&4&2&4&&&\\16\!:\!1&53&4&2&4&&&\\1\!:\!7&128&4&2&4&&&\\1\!:\!8&256&4&2&4&&&\\1\!:\!16&65536&4&2&4&8&16&20\\\hline \hline \end{array}}$

ASCII

 Example

    * Primal Functions, Primal Codes, Sort Parameters and Subtotals
    * ================================================================
    * Primal Function |       Primal Code   =   a | g q h | r | s | t
    * ================================================================
    * { }             |                         1 | 0 1 0 | 1 | 1 | 1
    * ================================================================
    * 1:1             |                         2 | 1 0 1 | 1 | 1 | 1
    * ================================================================
    * 2:1             |                         3 | 2 2 2 |   |   |
    * 1:2             |                         4 | 2 2 2 | 2 | 2 | 2
    * ================================================================
    * 1:1 2:1         |                         6 | 3 0 2 |   |   |
    * 2:2             |                         9 | 3 0 2 | 2 | 2 |
    * ----------------+---------------------------+-------+---+---+---
    * 3:1             |                         5 | 3 2 3 |   |   |
    * 4:1             |                         7 | 3 2 3 |   |   |
    * 1:3             |                         8 | 3 2 3 |   |   |
    * 1:4             |                        16 | 3 2 3 | 4 | 4 | 6
    * ================================================================
    * 1:2 2:1         |                        12 | 4 0 2 |   |   |
    * 1:1 2:2         |                        18 | 4 0 2 | 2 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 1:1 3:1         |                        10 | 4 0 3 |   |   |
    * 1:1 4:1         |                        14 | 4 0 3 | 2 | 4 |
    * ----------------+---------------------------+-------+---+---+---
    * 6:1             |                        13 | 4 2 3 |   |   |
    * 9:1             |                        23 | 4 2 3 |   |   |
    * 3:2             |                        25 | 4 2 3 |   |   |
    * 2:3             |                        27 | 4 2 3 |   |   |
    * 4:2             |                        49 | 4 2 3 |   |   |
    * 1:6             |                        64 | 4 2 3 |   |   |
    * 2:4             |                        81 | 4 2 3 |   |   |
    * 1:9             |                       512 | 4 2 3 | 8 |   |
    * ----------------+---------------------------+-------+---+---+---
    * 5:1             |                        11 | 4 2 4 |   |   |
    * 7:1             |                        17 | 4 2 4 |   |   |
    * 8:1             |                        19 | 4 2 4 |   |   |
    * 1:5             |                        32 | 4 2 4 |   |   |
    * 16:1            |                        53 | 4 2 4 |   |   |
    * 1:7             |                       128 | 4 2 4 |   |   |
    * 1:8             |                       256 | 4 2 4 |   |   |
    * 1:16            |                     65536 | 4 2 4 | 8 |16 |20
    * ================================================================
    * a = this sequence
    * g = rote weight in gammas = A062537
    * q = primal code character = A108352
    * h = rote height in gammas = A109301
    * r = number in (g,q,h) set = A113200
    * s = count in (g, q) class = A112869
    * t = count in weight class = A061396

A113200

Tetrahedron T(g, q, h) = number of rotes of weight g, quench q, height h.

TeX Array

${\begin{array}{l|*{10}{r}}\hline \\g\backslash (q,h)&(1,0)&(0,1)&(0,2)&&(0,3)&&&(0,4)&&\\&&&&(2,2)&&(2,3)&&&(2,4)&(2,5)\\&&&&&&&(3,3)&&&\\\hline \hline 0&1~\\\hline 1&&1~\\\hline 2&&&&2~\\\hline 3&&&2~&&\\&&&&&&4~\\\hline 4&&&2~&&2~&&\\&&&&&&8~&&&8~\\\hline 5&&&1~&&8~&&&4~&28~&16~\\&&&&&&12~&&&&\\&&&&&&&4~&&&\\\hline \end{array}}$

ASCII

 Example

    * Table T(g, q, h), omitting empty cells, starts out as follows:
    * --------+------------------------------------------------------------
    * g\(q,h) | (1,0) (0,1) (0,2)       (0,3)             (0,4)            
    *         |                   (2,2)       (2,3)             (2,4) (2,5)
    *         |                                     (3,3)                  
    * ========+============================================================
    * 0       |   1                                                        
    * --------+------------------------------------------------------------
    * 1       |         1                                                  
    * --------+------------------------------------------------------------
    * 2       |                     2                                      
    * --------+------------------------------------------------------------
    * 3       |               2                                            
    * 3       |                                 4                          
    * --------+------------------------------------------------------------
    * 4       |               2           2                                
    * 4       |                                 8                 8        
    * --------+------------------------------------------------------------
    * 5       |               1           8                 4              
    * 5       |                                12                28    16  
    * 5       |                                       4                    
    * --------+------------------------------------------------------------
    * Row sums = A112869. Horizontal section sums = A061396.

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16

User:Jon Awbrey/WORKSPACE

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Primal Codes of Finite Partial Functions on Positive Integers

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TeX Array

									1		1
								2		1		2
							3		1		1		3
						4		1		2		1		4
					5		1		3		1		1		5
				6		1		1		1		4		1		6
			7		1		5		2		9		1		1		7
		8		1		6		1		1		1		2		1		8
	9		1		7		1		25		1		3		1		1		9
10		1		1		1		36		1		2		1		8		1		10

															1		1
														2		·		2
													3		·		·		3
												4		·		2		·		4
											5		·		3		1		·		5
										6		·		1		1		4		·		6
									7		·		5		2		9		1		·		7
								8		·		6		1		1		1		2		·		8
							9		·		7		1		25		1		3		1		·		9
						10		·		1		1		36		1		2		1		8		·		10
					11		·		1		1		49		1		5		1		27		1		·		11
				12		·		10		3		1		1		6		1		1		1		2		·		12
			13		·		11		1		1		2		7		1		125		4		3		1		·		13
		14		·		3		1		100		1		1		1		216		1		1		1		4		·		14
	15		·		13		2		121		1		3		1		343		1		5		1		9		1		·		15
16		·		14		1		9		1		10		1		1		1		6		1		2		1		2		·		16

															1		1
														2		1		2
													3		2		1		3
												4		3		2		1		4
											5		4		1		2		1		5
										6		5		1		1		2		1		6
									7		6		1		1		1		2		1		7
								8		7		6		1		1		1		2		1		8
							9		8		1		6		1		1		1		2		1		9
						10		9		1		1		6		1		1		1		2		1		10
					11		10		9		1		1		6		1		1		1		2		1		11
				12		11		10		9		1		1		6		1		1		1		2		1		12
			13		12		1		10		9		1		1		6		1		1		1		2		1		13
		14		13		18		1		10		9		1		1		6		1		1		1		2		1		14
	15		14		1		12		1		10		9		1		1		6		1		1		1		2		1		15
16		15		14		1		18		1		10		9		1		1		6		1		1		1		2		1		16