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|[[Walsh permutation]]; [[Walsh permutation; bit permutation|bit permutation]] [[File:Rdrup.svg|link=]]
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Finite permutations ''p'' and corresponding bit permutations ''P'' are composed in opposite directions:
<math>p_a * p_b * ... = p_x ~~~~ \Leftrightarrow ~~~~ ... * P_b * P_a = P_x</math>
__TOC__
==p<sub>1</sub> * p<sub>8</sub>==
<math>p_1 * p_8 = p_9 ~~~~ \Leftrightarrow ~~~~ P_8 * P_1 = P_9</math>
<source lang="matlab">
rowp1=p1 = [0 1 0 0;
1 0 0 0;
0 0 1 0;
0 0 0 1];
rowp1*p1 * [1:4]'
ans =
2
1
3
4
rowp8=p8 = [1 0 0 0;
0 0 0 1;
0 1 0 0;
0 0 1 0];
rowp8*p8 * [1:4]'
ans =
1
4
2
3
rowp1*rowp8*p1 * p8 * [1:4]'
ans =
4
1
2
3
% The result is p9.
% p1 after p8 = p9
ROWP1=P1 = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1];
ROWP1*P1 * [0:15]'
ans =
0
2
1
3
4
6
5
7
8
10
9
11
12
14
13
15
ROWP8=P8 = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1];
ROWP8*P8 * [0:15]'
ans =
0
1
4
5
8
9
12
13
2
3
6
7
10
11
14
15
ROWP1*ROWP8*P1 * P8 * [0:15]'
ans =
0
4
1
5
8
12
9
13
2
6
3
7
10
14
11
15
% The result is not P9 but P10, the inverse permutation of P9.
% The row permutation matrices of the big permutations have to be multiplied like column permutation matrices from right to left
% to get the result corresponding to the small permutations:
ROWP8*ROWP1*P8 * P1 * [0:15]'
ans =
0
2
4
6
8
10
12
14
1
3
5
7
9
11
13
15
% The result is P9.
% P1 before P8 = P9
</source>
==p<sub>1</sub> * p<sub>3</sub> * p<sub>8</sub>==
<math>p_1 * p_3 * p_8 = p_6 ~~~~ \Leftrightarrow ~~~~ P_8 * P_3 * P_1 = P_6</math>
<source lang="matlab">
rowp1=p1 = [0 1 0 0;
1 0 0 0;
0 0 1 0;
0 0 0 1];
rowp3=p3 = [0 0 1 0;
1 0 0 0;
0 1 0 0;
0 0 0 1];
rowp8=p8 = [1 0 0 0;
0 0 0 1;
0 1 0 0;
0 0 1 0];
ROWP1=P1 = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1];
ROWP3=P3 = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1];
ROWP8=P8 = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0;
0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1];
rowp1*rowp3*rowp8*p1 * p3 * p8 * [1:4]'
ans =
1
2
4
3
% The result is p6.
% p1 after p3 after p8 = p6
ROWP8*ROWP3*ROWP1*P8 * P3 * P1 * [0:15]'
ans =
0
1
2
3
8
9
10
11
4
5
6
7
12
13
14
15
% The result is P6.
% P1 before P3 before P8 = P6
</source>
==p<sub>8</sub> * p<sub>3</sub> * p<sub>1</sub>==
<math>p_8 * p_3 * p_1 = p_{22} ~~~~ \Leftrightarrow ~~~~ P_1 * P_3 * P_8 = P_{22}</math>
<source lang="matlab">
rowp8*rowp3*rowp1*p8 * p3 * p1 * [1:4]'
ans =
3
4
2
1
% The result is p22.
% p8 after p3 after p1 = p22
ROWP1*ROWP3*ROWP8*P1 * P3 * P8 * [0:15]'
ans =
0
8
4
12
1
9
5
13
2
10
6
14
3
11
7
15
% The result is P22.
% P8 before P3 before P1 = P22
</source>
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[[Category:Walsh permutation]]
[[Category:Walsh permutation]]
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