3-bit Walsh permutation; table

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3-bit Walsh permutation Rdrup.svg
Array of 2-element subsets of an 8-element set
(compare Sloane'sA018900 as a square array)

The sortable table shows all 168 3-bit Walsh permutations. These are it's columns:

CV Compression vector
CM Compression matrix
D Real determinant of the compression matrix (+ stands for 1, − for −1)
Σ Number of ones in the compression matrix
# Index number of the finite permutation (compare Sloane'sA195664)
Permutation Walsh permutations of numbers 0,...,7
Inversion Vector Inversion vector (reflected factorial number) showing the number of bigger elements to the left of each element
IN Inversion number (sum of the inversion vector)
IS Shows which 2-cycles are part of the inversion set. Compare file on the right.
PM Permutation matrix (to permute rows, e.g. the elements of a column vector)
FPC Fano plane collineation
Cy Partition representing the cycle structure of the permutation (compare Sloane'sA198380)
14a and 14b are conjugacy classes (see here), so the indexes are information beyond the cycle structure.
I Sorting this column will show inverse permutations next to each other.


Plain text tables: Compression vectors, Permutations

Code of the binary matrices can be copied from their gray and red representations.


CV   CM   D Σ # Permutation
Inversion Vector
IN IS PM FPC Cy I
1 2 4

[1,0,0;
0,1,0;
0,0,1]

+ 3 0 0 1 2 3   4 5 6 7
0 0 0 0   0 0 0 0
 0

[0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;

0,0,0,0, 1,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,0,0,1]

Fanoperm124.svg &1   0 &124
7 2 4

[1,0,0;
1,1,0;
1,0,1]

+ 5 126 0 1 3 2   5 4 6 7
0 0 0 1   0 1 0 0
 2

[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;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,1,0, 0,0,0,0;

0,0,0,0, 0,1,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,0,0,1]

Fanoperm724.svg &2   3   (2+2)> &724
1 4 2

[1,0,0;
0,0,1;
0,1,0]

3 288 0 1 4 5   2 3 6 7
0 0 0 0   2 2 0 0
 4

[0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,1,1,0,0,0,0;
1,1,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,1,0,0;

0,0,1,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,0,0,1]

Fanoperm142.svg &2   3   (2+2)> &142
7 4 2

[1,0,0;
1,0,1;
1,1,0]

5 414 0 1 5 4   3 2 6 7
0 0 0 1   2 3 0 0
 6

[0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
1,1,1,0,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 1,0,0,0;

0,0,0,1, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,0,0,1]

Fanoperm742.svg &2   3   (2+2)> &742
2 1 4

[0,1,0;
1,0,0;
0,0,1]

3 722 0 2 1 3   4 6 5 7
0 0 1 0   0 0 1 0
 2

[0,0,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;

0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm214.svg &2   3   (2+2)> &214
2 7 4

[0,1,0;
1,1,0;
0,1,1]

5 852 0 2 3 1   6 4 5 7
0 0 0 2   0 1 1 0
 4

[0,0,0,0,0,0,0;
0,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,1,0,0, 0,0,0,0;

0,0,0,0, 0,0,1,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm274.svg &4   9   (3+3)> &274
4 1 2

[0,1,0;
0,0,1;
1,0,0]

+ 3 1032 0 2 4 6   1 3 5 7
0 0 0 0   3 2 1 0
 6

[0,0,0,0,0,0,0;
0,0,1,0,0,0,0;
0,1,1,0,0,0,0;
1,1,1,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,1,0;

0,1,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm412.svg &4   9   (3+3)> &241
4 7 2

[0,1,0;
0,1,1;
1,1,0]

+ 5 1254 0 2 6 4   3 1 5 7
0 0 0 1   2 4 1 0
 8

[0,0,0,0,0,0,0;
0,0,0,1,0,0,0;
1,1,1,1,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 1,0,0,0;

0,0,0,1, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm472.svg &3   8   (4+2)> &472
7 1 4

[1,1,0;
1,0,0;
1,0,1]

5 1448 0 3 1 2   5 6 4 7
0 0 1 1   0 0 2 0
 4

[0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;

0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm714.svg &4   9   (3+3)> &274
1 7 4

[1,1,0;
0,1,0;
0,1,1]

+ 5 1574 0 3 2 1   6 5 4 7
0 0 1 2   0 1 2 0
 6

[0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;

0,0,0,0, 0,0,1,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm174.svg &2   3   (2+2)> &174
7 1 2

[1,1,0;
1,0,1;
1,0,0]

+ 5 1872 0 3 5 6   1 2 4 7
0 0 0 0   3 3 2 0
 8

[0,0,0,0,0,0,0;
0,0,1,1,0,0,0;
0,1,1,1,0,0,0;
1,1,1,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 0,0,1,0;

0,1,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm712.svg &3   8   (4+2)> &247
1 7 2

[1,1,0;
0,1,1;
0,1,0]

5 1998 0 3 6 5   2 1 4 7
0 0 0 1   3 4 2 0
10

[0,0,0,0,0,0,0;
0,0,1,1,0,0,0;
1,1,1,1,0,0,0;
1,1,1,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 0,1,0,0;

0,0,1,0, 0,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm172.svg &4   9   (3+3)> &147
2 4 1

[0,0,1;
1,0,0;
0,1,0]

+ 3 2210 0 4 1 5   2 6 3 7
0 0 1 0   2 0 3 0
 6

[0,0,0,0,0,0,0;
1,0,1,0,1,0,0;
0,0,0,0,0,0,0;
1,0,1,0,0,0,0;
0,0,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;

0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm241.svg &4   9   (3+3)> &241
4 2 1

[0,0,1;
0,1,0;
1,0,0]

3 2354 0 4 2 6   1 5 3 7
0 0 1 0   3 1 3 0
 8

[0,0,0,0,0,0,0;
1,0,1,0,1,0,0;
0,1,0,0,0,0,0;
1,1,1,0,0,0,0;
0,0,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;

0,1,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm421.svg &2   3   (2+2)> &421
2 4 7

[0,0,1;
1,0,1;
0,1,1]

+ 5 2532 0 4 5 1   6 2 3 7
0 0 0 2   0 3 3 0
 8

[0,0,0,0,0,0,0;
0,1,0,1,1,0,0;
1,0,1,1,0,0,0;
0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
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, 1,0,0,0;
0,0,0,0, 0,1,0,0;
0,1,0,0, 0,0,0,0;

0,0,0,0, 0,0,1,0;
0,0,1,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm247.svg &3   8   (4+2)> &247
4 2 7

[0,0,1;
0,1,1;
1,0,1]

5 2676 0 4 6 2   5 1 3 7
0 0 0 2   1 4 3 0
10

[0,0,0,0,0,0,0;
0,1,0,1,1,0,0;
1,1,1,1,0,0,0;
0,1,0,0,0,0,0;
1,1,0,0,0,0,0;
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, 1,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,1,0, 0,0,0,0;

0,0,0,0, 0,1,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm427.svg &4   9   (3+3)> &427
7 4 1

[1,0,1;
1,0,0;
1,1,0]

+ 5 2936 0 5 1 4   3 6 2 7
0 0 1 1   2 0 4 0
 8

[0,0,0,0,0,0,0;
1,1,1,0,1,0,0;
0,0,0,0,0,0,0;
1,0,1,0,0,0,0;
0,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;

0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm741.svg &3   8   (4+2)> &472
7 2 1

[1,0,1;
1,1,0;
1,0,0]

5 3194 0 5 3 6   1 4 2 7
0 0 1 0   3 2 4 0
10

[0,0,0,0,0,0,0;
1,0,1,1,1,0,0;
0,1,0,1,0,0,0;
1,1,1,0,0,0,0;
0,0,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,0,1,0;

0,1,0,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm721.svg &4   9   (3+3)> &427
1 4 7

[1,0,1;
0,0,1;
0,1,1]

5 3254 0 5 4 1   6 3 2 7
0 0 1 2   0 3 4 0
10

[0,0,0,0,0,0,0;
1,1,0,1,1,0,0;
1,0,1,1,0,0,0;
0,0,0,0,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,0,0,0, 1,0,0,0;
0,1,0,0, 0,0,0,0;

0,0,0,0, 0,0,1,0;
0,0,0,1, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm147.svg &4   9   (3+3)> &147
1 2 7

[1,0,1;
0,1,1;
0,0,1]

+ 5 3420 0 5 6 3   4 1 2 7
0 0 0 2   2 4 4 0
12

[0,0,0,0,0,0,0;
0,1,1,1,1,0,0;
1,1,1,1,0,0,0;
0,1,1,0,0,0,0;
1,1,0,0,0,0,0;
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, 0,1,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,1, 0,0,0,0;

0,0,0,0, 1,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm127.svg &2   3   (2+2)> &127
4 7 1

[0,1,1;
0,1,0;
1,1,0]

5 3776 0 6 2 4   3 5 1 7
0 0 1 1   2 1 5 0
10

[0,0,0,0,0,0,0;
1,1,1,1,1,0,0;
0,0,0,1,0,0,0;
1,0,1,0,0,0,0;
0,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,1,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;

0,0,0,1, 0,0,0,0;
0,0,0,0, 0,1,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm471.svg &2   3   (2+2)> &471
2 7 1

[0,1,1;
1,1,0;
0,1,0]

+ 5 3920 0 6 3 5   2 4 1 7
0 0 1 1   3 2 5 0
12

[0,0,0,0,0,0,0;
1,1,1,1,1,0,0;
0,1,0,1,0,0,0;
1,1,1,0,0,0,0;
0,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,1, 0,0,0,0;
0,0,0,0, 0,1,0,0;

0,0,1,0, 0,0,0,0;
0,0,0,0, 1,0,0,0;
0,1,0,0, 0,0,0,0;
0,0,0,0, 0,0,0,1]

Fanoperm271.svg &3   8   (4+2)> &271
4 1 7

[0,1,1;
0,0,1;
1,0,1]

+ 5 3998 0 6 4 2   5 3 1 7
0 0 1 2   1 3 5 0
12

[0,0,0,0,0,0,0;
1,1,1,1,1,0,0;
1,0,1,1,0,0,0;
0,0,1,0,0,0,0;
1,1,0,0,0,0,0;
1,0,0,0,0,0,0;
0,0,0,0,0,0,0]

[1,0,0,0, 0,0,0,0;
0,0,0,0, 0,0,1,0;
0,0,0,0, 1,0,0,0;
0,0,1,0, 0,0,0,0;

0,0,0,0, 0,1,0,0;
0,0,0,1, 0,0,0,0;
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Fanoperm217.svg &2   3   (2+2)> &217
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Fanoperm324.svg &2   3   (2+2)> &324
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Fanoperm524.svg &2   3   (2+2)> &524
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Fanoperm342.svg &3   8   (4+2)> &342
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Fanoperm542.svg &3   8   (4+2)> &342
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Fanoperm314.svg &4   9   (3+3)> &234
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Fanoperm574.svg &3   8   (4+2)> &534
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Fanoperm512.svg &5  14a   (7)> &243
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Fanoperm372.svg &6  14b   (7)> &372
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Fanoperm614.svg &3   8   (4+2)> &254
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Fanoperm674.svg &4   9   (3+3)> &354
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Fanoperm612.svg &6  14b   (7)> &245
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Fanoperm672.svg &5  14a   (7)> &345
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Fanoperm341.svg &5  14a   (7)> &341
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Fanoperm521.svg &4   9   (3+3)> &425
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Fanoperm547.svg &6  14b   (7)> &352
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Fanoperm327.svg &3   8   (4+2)> &325
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Fanoperm641.svg &6  14b   (7)> &432
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Fanoperm621.svg &3   8   (4+2)> &423
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Fanoperm647.svg &5  14a   (7)> &532
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Fanoperm627.svg &4   9   (3+3)> &523
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Fanoperm371.svg &4   9   (3+3)> &371
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Fanoperm517.svg &4   9   (3+3)> &253
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Fanoperm254.svg &3   8   (4+2)> &254
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Fanoperm432.svg &5  14a   (7)> &432
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Fanoperm452.svg &6  14b   (7)> &341
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Fanoperm134.svg &2   3   (2+2)> &134
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Fanoperm754.svg &4   9   (3+3)> &634
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Fanoperm732.svg &4   9   (3+3)> &643
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Fanoperm152.svg &3   8   (4+2)> &143
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Fanoperm164.svg &2   3   (2+2)> &164
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Fanoperm764.svg &3   8   (4+2)> &364
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Fanoperm162.svg &4   9   (3+3)> &146
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Fanoperm762.svg &4   9   (3+3)> &346
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Fanoperm431.svg &3   8   (4+2)> &431
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Fanoperm251.svg &6  14b   (7)> &251
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Fanoperm237.svg &4   9   (3+3)> &237
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