User:Hubby56/sandbox

Back to basics in CS

Footnotes removal
by AMS and MIT footnotes are not welcomed in a mathematics article.

AMS
In this connection it might be remarked that an excessive number of footnotes sometimes gives the undesirable impression that the paper is being "written in the footnotes." Careful organization, however, always makes it possible to reduce the number and length of footnotes, even to eliminate them entirely.

A Manual for Authors of Mathematical Papers -A Manual for Authors of Mathematical Papers - American Mathematical Society. page 4

http://www.ams.org/journals/bull/1943-49-03/S0002-9904-1943-07884-6/S0002-9904-1943-07884-6.pdf

MIT
The citation is treated somewhat like a parenthetical remark within a sentence, but the reason for the citation must be immediately apparent. Footnotes are not used;

Writing a Math Phase Two Paper - MIT page 5

http://web.mit.edu/mathp2/www/piil.html

Colored permutation
Take an uncolored permutation, e.g. $$\sigma = (A)(B)(C)(D,E,F)$$. Suppose we can color permutations i.e. elements in the same cycle have the same color.

Then, for each colored $$\sigma_c$$, e.g. $$ \sigma_c = \color {red} (A) \color {green} (B) \color {blue} (C) \color {green} (D, E, F)$$ we associate  a colored partition $$ |\ \color {red}   A}\ |\ {\color {green}  B, D, E, F }\ |\ {\color {blue} C\ | $$  that is fixed by $$\sigma = (A)(B)(C)(D,E,F)$$.

Different colorings of $$\sigma = (A)(B)(C)(D,E,F)$$ produces different colored partitions.

Reversely, given a colored partition eg. $$ |\ \color {red}   A,B\ |\ \color {green}  C, D, E, F \ |$$  we observe that, to be fixed by fixed by $$\sigma$$, the color must be constant over a cycle.

Thus, there is a coloring $$\sigma_c = \color {red} (A)(B) \color {green}(C)(D,E,F)$$ that produces the given partition.

In conclusion, there is a one-to-one correspondence between colorings of a permutation and colored partitions fixed by that permutation.

With you permission I will continue the approach via colored permutations introduced to another section below. Take now $$ \sum \ of \ all \ colored \ permutations \ like \ would \ be \ \color {red}c_1 \color {green} c_1 \color {blue}c_1\color {green}c_3 $$ and re-groupe it like

$$ \sum of \ colored  \ permutations \ like \ ( \color{red}c_1 \color{black} + \color{green} c_1 \color{black} + \color {blue} c_1 \color{black} ) \cdot ( \color{red}c_1 \color{black} + \color{green} c_1 \color{black} + \color {blue} c_1 \color{black} ) \cdot ( \color{red}c_1 \color{black} + \color{green} c_1 \color{black} + \color {blue} c_1 \color{black} ) \cdot ( \color{red}c_3 \color{black} + \color{green} c_3 \color{black} + \color {blue} c_3 \color{black} ) $$

We build now a generating polynomial by formally replacing in the above expression colored cycles with "counters"

$$ \sum of \ factors \ like \ ( r + g  +  b ) \cdot ( r + g  +  b ) \cdot ( r + g +  b ) \cdot ( r^3 + g^3  +  b^3 ) $$

the exponents of the "counters" r, g, b are not counting only the length of cycles but also indicates the distribution of colors on the colored cube.

For our cube, the generating polynomial has 28 terms. Suppose we want to know the number of colorings depending on one color - the other not being specified. We take $$g=b=1$$ (the so called zero weight) and we obtain :

$$r^6 + 2r^5 + 6r^4 + 10r^3 + 16r^2 + 12r + 10$$

10 stands here for 10 different colorings using only g and b.

two rectangles for each square, three squares for each quadrangle
$$ {1 \over 24} (a_1^4 + 6.a_1^2.a_2 + 8.a_1a_3 + 3 a_2^2 + 6.a^4) $$

REFD
BLABLA