3-bit Walsh permutation/arrow patterns

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matrix sums from 3 to 6
fixed points

This table shows the 24 arrow patterns of the 168 3-bit Walsh permutations (as Fano plane collineations).

The typical shape of a collection is a hexagon, showing the two rotations and three reflections of some initial pattern.
The one pattern with rotational symmetry is shown as a pair. Patterns with mirror symmetry are shown as triangles.
Where the hexagon or triangle does not contain the inverse permutations, two nested polygons are shown.
These are the types of collections shown below:

  • neutral permutation
  • 1 pair (of rotations)

  • 5 single triangles
  • 1 double triangle

  • 8 single hexagons
  • 8 double hexagons

In the column graph, the shown permutations are highlighted in the respective component of the neighbor graph. All except 9 are in the cube-like clusters with 25 vertices. Apart from the center, there are four types of vertices, which are marked with suits.

Each permutation has a set of fixed points.

  • The neutral permutation has only fixed points.
  • 7-cycles have none.

  • For cycle type 2+2 they form a line of the Fano plane.

  • Cycle types 2+4 and 3+3 have single fixed points.

The points can be characterized by their binary weight, and the lines by the weights of their points. This property is in the column fixed.

size collection graph det sum cycles fixed
&1 1
Fanoperm124.svg

+ middle S3 Fano patterns in neighbor graph; positive middle star (124).svg

&1 + 3 &0 &1
&5 6

Walsh permutation 134 Fano.svg

Walsh permutation 164 Fano.svg

Walsh permutation 324 Fano.svg

Walsh permutation 126 Fano.svg

Walsh permutation 524 Fano.svg

Walsh permutation 125 Fano.svg


+ middle S4 Fano patterns in neighbor graph; positive middle spades (134).svg

&1 + 4 &2 22 &2 121
&3 3

Walsh permutation 127 Fano.svg

Walsh permutation 724 Fano.svg

Walsh permutation 174 Fano.svg


+ middle S5a F121 Fano patterns in neighbor graph; positive middle hearts 127.svg

&1 + 5a &2 22 &2 121
&3 3

Walsh permutation 564 Fano.svg

Walsh permutation 135 Fano.svg

Walsh permutation 326 Fano.svg


+ middle S5a F132 Fano patterns in neighbor graph; positive middle hearts 564.svg

&1 + 5a &2 22 &3 132
&6 6 * 2

Walsh permutation 137 Fano.svg

Walsh permutation 764 Fano.svg

Walsh permutation 136 Fano.svg

Walsh permutation 364 Fano.svg

Walsh permutation 327 Fano.svg

Walsh permutation 726 Fano.svg

Walsh permutation 325 Fano.svg

Walsh permutation 526 Fano.svg

Walsh permutation 534 Fano.svg

Walsh permutation 165 Fano.svg

Walsh permutation 574 Fano.svg

Walsh permutation 175 Fano.svg


outside: + middle S6 Fano patterns in neighbor graph; positive middle diamonds (137).svg inside: + middle S5b Fano patterns in neighbor graph; positive middle clubs (136).svg

&1 + 5b
6
&3 24 &5 1
&2 2

Walsh permutation 241 Fano.svg

Walsh permutation 412 Fano.svg

+ sides S3 Fano patterns in neighbor graph; positive sides star (241).svg

&1 + 3 &4 33 &7 3
&5 6

Walsh permutation 741 Fano.svg

Walsh permutation 417 Fano.svg

Walsh permutation 472 Fano.svg

Walsh permutation 271 Fano.svg

Walsh permutation 247 Fano.svg

Walsh permutation 712 Fano.svg


+ sides S5a F2 Fano patterns in neighbor graph; positive sides hearts 741.svg

&1 + 5a &3 24 &6 2
&5 6

Walsh permutation 263 Fano.svg

Walsh permutation 632 Fano.svg

Walsh permutation 513 Fano.svg

Walsh permutation 645 Fano.svg

Walsh permutation 351 Fano.svg

Walsh permutation 456 Fano.svg


+ sides S5a F3 Fano patterns in neighbor graph; positive sides hearts 263.svg

&1 + 5a &3 24 &7 3
&6 6 * 2

Walsh permutation 413 Fano.svg

Walsh permutation 641 Fano.svg

Walsh permutation 261 Fano.svg

Walsh permutation 432 Fano.svg

Walsh permutation 243 Fano.svg

Walsh permutation 612 Fano.svg

Walsh permutation 512 Fano.svg

Walsh permutation 245 Fano.svg

Walsh permutation 341 Fano.svg

Walsh permutation 416 Fano.svg

Walsh permutation 452 Fano.svg

Walsh permutation 251 Fano.svg


outside: + sides S4 C7b Fano patterns in neighbor graph; positive sides spades orange (413).svg inside: + sides S4 C7a Fano patterns in neighbor graph; positive sides spades yellow (261).svg

&1 + 4 &5 7 &8
&6 6 * 2

Walsh permutation 253 Fano.svg

Walsh permutation 652 Fano.svg

Walsh permutation 517 Fano.svg

Walsh permutation 745 Fano.svg

Walsh permutation 613 Fano.svg

Walsh permutation 643 Fano.svg

Walsh permutation 267 Fano.svg

Walsh permutation 732 Fano.svg

Walsh permutation 476 Fano.svg

Walsh permutation 371 Fano.svg

Walsh permutation 651 Fano.svg

Walsh permutation 453 Fano.svg


outside: + sides S5b C33 Fano patterns in neighbor graph; positive sides clubs white (253).svg inside: + sides S6 C33 Fano patterns in neighbor graph; positive sides diamonds white (517).svg

&1 + 5b
6
&4 33 &6 2
&6 6 * 2

Walsh permutation 713 Fano.svg

Walsh permutation 647 Fano.svg

Walsh permutation 265 Fano.svg

Walsh permutation 532 Fano.svg

Walsh permutation 273 Fano.svg

Walsh permutation 672 Fano.svg

Walsh permutation 516 Fano.svg

Walsh permutation 345 Fano.svg

Walsh permutation 361 Fano.svg

Walsh permutation 436 Fano.svg

Walsh permutation 457 Fano.svg

Walsh permutation 751 Fano.svg


outside: + sides S6 C7a Fano patterns in neighbor graph; positive sides diamonds yellow (713).svg inside: + sides S5b C7b Fano patterns in neighbor graph; positive sides clubs orange (265).svg

&1 + 5b
6
&5 7 &8
&6 6 * 2

Walsh permutation 763 Fano.svg

Walsh permutation 637 Fano.svg

Walsh permutation 375 Fano.svg

Walsh permutation 576 Fano.svg

Walsh permutation 573 Fano.svg

Walsh permutation 675 Fano.svg

Walsh permutation 736 Fano.svg

Walsh permutation 367 Fano.svg

Walsh permutation 765 Fano.svg

Walsh permutation 537 Fano.svg

Walsh permutation 357 Fano.svg

Walsh permutation 756 Fano.svg


outside: − matrix triangles Fano patterns in neighbor graph; negative matrix triangles (763).svg inside: + matrix triangles Fano patterns in neighbor graph; positive matrix triangles (375).svg

&2 ± 7 &5 7 &8
&4 3 * 2

Walsh permutation 567 Fano.svg

Walsh permutation 735 Fano.svg

Walsh permutation 657 Fano.svg

Walsh permutation 753 Fano.svg

Walsh permutation 673 Fano.svg

Walsh permutation 376 Fano.svg

outside: − matrix diagonal Fano patterns in neighbor graph; negative matrix diagonal (567).svg inside: + matrix diagonal Fano patterns in neighbor graph; positive matrix diagonal (657).svg

&2 ± 7 &3 24 &6 2
&3 3

Walsh permutation 214 Fano.svg

Walsh permutation 142 Fano.svg

Walsh permutation 421 Fano.svg


S3 Fano patterns in neighbor graph; negative star (214).svg

&3 3 &2 22 &3 132
&3 3

Walsh permutation 217 Fano.svg

Walsh permutation 742 Fano.svg

Walsh permutation 471 Fano.svg


S5a C22 F222 Fano patterns in neighbor graph; negative hearts red 217.svg

&3 5a &2 22 &4 222
&3 3

Walsh permutation 654 Fano.svg

Walsh permutation 153 Fano.svg

Walsh permutation 623 Fano.svg


S5a C22 F132 Fano patterns in neighbor graph; negative hearts red 654.svg

&3 5a &2 22 &3 132
&5 6

Walsh permutation 274 Fano.svg

Walsh permutation 172 Fano.svg

Walsh permutation 714 Fano.svg

Walsh permutation 147 Fano.svg

Walsh permutation 721 Fano.svg

Walsh permutation 427 Fano.svg


S5a C33 F1 Fano patterns in neighbor graph; negative hearts white 274.svg

&3 5a &4 33 &5 1
&5 6

Walsh permutation 236 Fano.svg

Walsh permutation 362 Fano.svg

Walsh permutation 315 Fano.svg

Walsh permutation 546 Fano.svg

Walsh permutation 531 Fano.svg

Walsh permutation 465 Fano.svg


S5a C33 F3 Fano patterns in neighbor graph; negative hearts white 236.svg

&3 5a &4 33 &7 3
&5 6

Walsh permutation 234 Fano.svg

Walsh permutation 162 Fano.svg

Walsh permutation 314 Fano.svg

Walsh permutation 146 Fano.svg

Walsh permutation 521 Fano.svg

Walsh permutation 425 Fano.svg


S4 C33 Fano patterns in neighbor graph; negative spades white (234).svg

&3 4 &4 33 &5 1
&5 6

Walsh permutation 215 Fano.svg

Walsh permutation 542 Fano.svg

Walsh permutation 216 Fano.svg

Walsh permutation 342 Fano.svg

Walsh permutation 461 Fano.svg

Walsh permutation 431 Fano.svg


S4 C24 F2 Fano patterns in neighbor graph; negative spades green 215.svg

&3 4 &3 24 &6 2
&5 6

Walsh permutation 254 Fano.svg

Walsh permutation 152 Fano.svg

Walsh permutation 614 Fano.svg

Walsh permutation 143 Fano.svg

Walsh permutation 621 Fano.svg

Walsh permutation 423 Fano.svg


S4 C24 F1 Fano patterns in neighbor graph; negative spades green 254.svg

&3 4 &3 24 &5 1
&6 6 * 2

Walsh permutation 237 Fano.svg

Walsh permutation 762 Fano.svg

Walsh permutation 316 Fano.svg

Walsh permutation 346 Fano.svg

Walsh permutation 317 Fano.svg

Walsh permutation 746 Fano.svg

Walsh permutation 235 Fano.svg

Walsh permutation 562 Fano.svg

Walsh permutation 435 Fano.svg

Walsh permutation 561 Fano.svg

Walsh permutation 571 Fano.svg

Walsh permutation 475 Fano.svg


outside: S6 C33 F2 Fano patterns in neighbor graph; negative diamonds white 237.svg inside: S5b C33 F2 Fano patterns in neighbor graph; negative clubs white 316.svg

&3 5b
6
&4 33 &6 2
&6 6 * 2

Walsh permutation 163 Fano.svg

Walsh permutation 634 Fano.svg

Walsh permutation 157 Fano.svg

Walsh permutation 754 Fano.svg

Walsh permutation 523 Fano.svg

Walsh permutation 625 Fano.svg

Walsh permutation 627 Fano.svg

Walsh permutation 723 Fano.svg

Walsh permutation 674 Fano.svg

Walsh permutation 173 Fano.svg

Walsh permutation 354 Fano.svg

Walsh permutation 156 Fano.svg


outside: S5b C33 F1 Fano patterns in neighbor graph; negative clubs white 163.svg inside: S6 C33 F1 Fano patterns in neighbor graph; negative diamonds white 157.svg

&3 5b
6
&4 33 &5 1
&6 6 * 2

Walsh permutation 463 Fano.svg

Walsh permutation 631 Fano.svg

Walsh permutation 731 Fano.svg

Walsh permutation 467 Fano.svg

Walsh permutation 543 Fano.svg

Walsh permutation 615 Fano.svg

Walsh permutation 372 Fano.svg

Walsh permutation 276 Fano.svg

Walsh permutation 547 Fano.svg

Walsh permutation 715 Fano.svg

Walsh permutation 352 Fano.svg

Walsh permutation 256 Fano.svg


outside: S5b C7a Fano patterns in neighbor graph; negative clubs yellow (463).svg inside: S6 C7b Fano patterns in neighbor graph; negative diamonds orange (731).svg

&3 5b
6
&5 7 &8