12,080
edits
Changes
User:Jon Awbrey/SEQUENCES (view source)
Revision as of 21:48, 16 December 2009
, 21:48, 16 December 2009add user workspace
==A061396==
* [http://oeis.org/wiki/A061396 A061396]
===Plain Wiki Table===
<br>
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
|- style="height:48px; background:#f0f0ff"
| <math>\text{Integer}\!</math>
| <math>\text{Factorization}\!</math>
| <math>\text{Notation}\!</math>
| <math>\text{Riff Digraph}\!</math>
| <math>\text{Rote Graph}\!</math>
| <math>\text{Traversal}\!</math>
|- style="height:48px"
| <math>1\!</math>
| <math>1\!</math>
|
|
| [[Image:Rooted Node Big.jpg|20px]]
|
|-
| <math>2\!</math>
| <math>\text{p}_1^1\!</math>
| <math>\text{p}\!</math>
| [[Image:Rooted Node Big.jpg|20px]]
| [[Image:Rote 2 Big.jpg|40px]]
| <math>((~))</math>
|-
| <math>3\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p}_\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 3 Big.jpg|40px]]
| <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|-
| <math>5\!</math>
|
<math>\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[6pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 5 Big.jpg|40px]]
| <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
|-
| <math>7\!</math>
|
<math>\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[6pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[6pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[6pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
<br>
===Nested Wiki Table===
<br>
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
|- style="height:50px; background:#f0f0ff"
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="19%" | <math>\text{Factorization}\!</math>
| width="14%" | <math>\text{Notation}\!</math>
| width="19%" | <math>\text{Riff Digraph}\!</math>
| width="19%" | <math>\text{Rote Graph}\!</math>
| width="19%" | <math>\text{Traversal}\!</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="19%" | <math>1\!</math>
| width="14%" |
| width="19%" |
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
| width="19%" |
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>2\!</math>
| width="19%" | <math>\text{p}_1^1\!</math>
| width="14%" | <math>\text{p}\!</math>
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
| width="19%" | <math>((~))</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="19%" |
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| width="14%" | <math>\text{p}_\text{p}\!</math>
| width="19%" | <math>\cdots</math>
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
| width="19%" | <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="19%" |
<math>\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}</math>
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="19%" | <math>\cdots</math>
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
| width="19%" | <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
|-
| <math>7\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
|}
<br>
===Old ASCII Version===
<pre>
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;
---------------------------------------------------
</pre>
==Example of a Raw HTML Table==
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<tr>
<td style="border-bottom:1px solid black"><math>\text{Mnemonic}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Category}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Classical Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Alternate Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Symmetric Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Operator}\!</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{E}
\\
\mathrm{Exclusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{A}
\\
\mathrm{Absolute}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{00}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{O}
\\
\mathrm{Obtrusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{10}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{I}
\\
\mathrm{Indefinite}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\ell_{11}</math></td></tr>
</table>
* [http://oeis.org/wiki/A061396 A061396]
===Plain Wiki Table===
<br>
{| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%"
|+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
|- style="height:48px; background:#f0f0ff"
| <math>\text{Integer}\!</math>
| <math>\text{Factorization}\!</math>
| <math>\text{Notation}\!</math>
| <math>\text{Riff Digraph}\!</math>
| <math>\text{Rote Graph}\!</math>
| <math>\text{Traversal}\!</math>
|- style="height:48px"
| <math>1\!</math>
| <math>1\!</math>
|
|
| [[Image:Rooted Node Big.jpg|20px]]
|
|-
| <math>2\!</math>
| <math>\text{p}_1^1\!</math>
| <math>\text{p}\!</math>
| [[Image:Rooted Node Big.jpg|20px]]
| [[Image:Rote 2 Big.jpg|40px]]
| <math>((~))</math>
|-
| <math>3\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p}_\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 3 Big.jpg|40px]]
| <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|-
| <math>5\!</math>
|
<math>\begin{array}{lll}
\text{p}_3^1
& = & \text{p}_{\text{p}_2^1}^1
\\[6pt]
& = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1
\end{array}</math>
| <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 5 Big.jpg|40px]]
| <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
|-
| <math>7\!</math>
|
<math>\begin{array}{lll}
\text{p}_4^1
& = & \text{p}_{\text{p}_1^2}^1
\\[6pt]
& = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1
\end{array}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^3
& = & \text{p}_1^{\text{p}_2^1}
\\[6pt]
& = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1}
\end{array}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^4
& = & \text{p}_1^{\text{p}_1^2}
\\[6pt]
& = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}}
\end{array}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
<br>
===Nested Wiki Table===
<br>
{| align="center" border="1" width="96%"
|+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math>
|- style="height:50px; background:#f0f0ff"
|
{| cellpadding="12" style="background:#f0f0ff; text-align:center; width:100%"
| width="10%" | <math>\text{Integer}\!</math>
| width="19%" | <math>\text{Factorization}\!</math>
| width="14%" | <math>\text{Notation}\!</math>
| width="19%" | <math>\text{Riff Digraph}\!</math>
| width="19%" | <math>\text{Rote Graph}\!</math>
| width="19%" | <math>\text{Traversal}\!</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>1\!</math>
| width="19%" | <math>1\!</math>
| width="14%" |
| width="19%" |
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
| width="19%" |
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>2\!</math>
| width="19%" | <math>\text{p}_1^1\!</math>
| width="14%" | <math>\text{p}\!</math>
| width="19%" | [[Image:Rooted Node Big.jpg|20px]]
| width="19%" | [[Image:Rote 2 Big.jpg|40px]]
| width="19%" | <math>((~))</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>3\!</math>
| width="19%" |
<math>\begin{array}{lll}
\text{p}_2^1 & = & \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| width="14%" | <math>\text{p}_\text{p}\!</math>
| width="19%" | <math>\cdots</math>
| width="19%" | [[Image:Rote 3 Big.jpg|40px]]
| width="19%" | <math>(((~))(~))</math>
|-
| <math>4\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^2 & = & \text{p}_1^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 4 Big.jpg|65px]]
| <math>((((~))))</math>
|}
|-
|
{| cellpadding="12" style="text-align:center; width:100%"
| width="10%" | <math>5\!</math>
| width="19%" |
<math>\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}</math>
| width="14%" | <math>\text{p}_{\text{p}_{\text{p}}}\!</math>
| width="19%" | <math>\cdots</math>
| width="19%" | [[Image:Rote 5 Big.jpg|40px]]
| width="19%" | <math>((((~))(~))(~))</math>
|-
| <math>6\!</math>
|
<math>\begin{array}{lll}
\text{p}_1^1 \text{p}_2^1
& = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^1
\end{array}</math>
| <math>\text{p} \text{p}_{\text{p}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 6 Big.jpg|80px]]
| <math>((~))(((~))(~))</math>
|-
| <math>7\!</math>
|
<math>\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}</math>
| <math>\text{p}_{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 7 Big.jpg|65px]]
| <math>(((((~))))(~))</math>
|-
| <math>8\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}_{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 8 Big.jpg|65px]]
| <math>(((((~))(~))))</math>
|-
| <math>9\!</math>
|
<math>\begin{array}{lll}
\text{p}_2^2
& = & \text{p}_{\text{p}_1^1}^{\text{p}_1^1}
\end{array}</math>
| <math>\text{p}_\text{p}^\text{p}\!</math>
| <math>\cdots</math>
| [[Image:Rote 9 Big.jpg|80px]]
| <math>(((~))(((~))))</math>
|-
| <math>16\!</math>
|
<math>\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}</math>
| <math>\text{p}^{\text{p}^{\text{p}}}\!</math>
| <math>\cdots</math>
| [[Image:Rote 16 Big.jpg|90px]]
| <math>((((((~))))))</math>
|}
|}
<br>
===Old ASCII Version===
<pre>
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;
---------------------------------------------------
</pre>
==Example of a Raw HTML Table==
<table align="center" cellpadding="4" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:90%">
<caption><font size="+2"><math>\text{Table 10.} ~~ \text{Relation of Quantifiers to Higher Order Propositions}</math></font></caption>
<tr>
<td style="border-bottom:1px solid black"><math>\text{Mnemonic}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Category}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Classical Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Alternate Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Symmetric Form}\!</math></td>
<td style="border-bottom:1px solid black"><math>\text{Operator}\!</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{E}
\\
\mathrm{Exclusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{All} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{No} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{11} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{A}
\\
\mathrm{Absolute}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black">
<math>\begin{matrix}
\mathrm{Universal}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ u ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{10} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{All} ~ v ~ \mathrm{is} ~ u</math></td>
<td><math>\mathrm{No} ~ v ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td><math>\texttt{(} \ell_{01} \texttt{)}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{All} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ u</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} v \texttt{)} ~ \mathrm{is} ~ \texttt{(} u \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\mathrm{No} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td style="border-bottom:1px solid black"><math>\texttt{(} \ell_{00} \texttt{)}</math></td></tr>
<tr>
<td> </td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{00}</math></td></tr>
<tr>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"> </td>
<td style="border-bottom:1px solid black"><math>\mathrm{Some} ~ \texttt{(} u \texttt{)} ~ \mathrm{is} ~ v</math></td>
<td style="border-bottom:1px solid black"><math>\ell_{01}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{O}
\\
\mathrm{Obtrusive}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Negative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ \texttt{(} v \texttt{)}</math></td>
<td><math>\ell_{10}</math></td></tr>
<tr>
<td><math>\begin{matrix}
\mathrm{I}
\\
\mathrm{Indefinite}
\end{matrix}</math></td>
<td><math>\begin{matrix}
\mathrm{Particular}
\\
\mathrm{Affirmative}
\end{matrix}</math></td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td> </td>
<td><math>\mathrm{Some} ~ u ~ \mathrm{is} ~ v</math></td>
<td><math>\ell_{11}</math></td></tr>
</table>