# Schoute permutation

* Schoute matrix
* rows: Schoute permutations
* columns: Schoute partitions
A Schoute permutation is a Walsh permutation corresponding to a permutation matrix. It can be seen as a permutation of hypercube vertices derived from a permutation of coordinate axes. In the signed case (see below) the axes can also change their directions.
A Schoute permutation or degree d has length 2$^d$. Like with all Walsh permutations, it makes sense to see them as periodic. The n-th Schoute permutation is row n of the number triangle  A195665.
The most well known Schoute permutations are the bit-reversal permutations. They are always the last row in a finite Schoute matrix, because the reversal is always the last among the finite permutations.
Also well known (but very forgettable) are those that simply sort the even before the odd numbers. (The corresponding finite permutations are the left shifts.)
| Schoute permutations | signed Schoute permutations |
examples with degree 4   (Python code) :
| n | n-th finite permutation | n-th Schoute permutation |
| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | [[0, 1, 2, 3], [1, 0, 2, 3], [0, 2, 1, 3], [2, 0, 1, 3], [1, 2, 0, 3], [2, 1, 0, 3], [0, 1, 3, 2], [1, 0, 3, 2], [0, 3, 1, 2], [3, 0, 1, 2], [1, 3, 0, 2], [3, 1, 0, 2], [0, 2, 3, 1], [2, 0, 3, 1], [0, 3, 2, 1], [3, 0, 2, 1], [2, 3, 0, 1], [3, 2, 0, 1], [1, 2, 3, 0], [2, 1, 3, 0], [1, 3, 2, 0], [3, 1, 2, 0], [2, 3, 1, 0], [3, 2, 1, 0]] | [[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [ 0, 2, 1, 3, 4, 6, 5, 7, 8, 10, 9, 11, 12, 14, 13, 15], [ 0, 1, 4, 5, 2, 3, 6, 7, 8, 9, 12, 13, 10, 11, 14, 15], [ 0, 4, 1, 5, 2, 6, 3, 7, 8, 12, 9, 13, 10, 14, 11, 15], [ 0, 2, 4, 6, 1, 3, 5, 7, 8, 10, 12, 14, 9, 11, 13, 15], [ 0, 4, 2, 6, 1, 5, 3, 7, 8, 12, 10, 14, 9, 13, 11, 15], [ 0, 1, 2, 3, 8, 9, 10, 11, 4, 5, 6, 7, 12, 13, 14, 15], [ 0, 2, 1, 3, 8, 10, 9, 11, 4, 6, 5, 7, 12, 14, 13, 15], [ 0, 1, 8, 9, 2, 3, 10, 11, 4, 5, 12, 13, 6, 7, 14, 15], [ 0, 8, 1, 9, 2, 10, 3, 11, 4, 12, 5, 13, 6, 14, 7, 15], [ 0, 2, 8, 10, 1, 3, 9, 11, 4, 6, 12, 14, 5, 7, 13, 15], [ 0, 8, 2, 10, 1, 9, 3, 11, 4, 12, 6, 14, 5, 13, 7, 15], [ 0, 1, 4, 5, 8, 9, 12, 13, 2, 3, 6, 7, 10, 11, 14, 15], [ 0, 4, 1, 5, 8, 12, 9, 13, 2, 6, 3, 7, 10, 14, 11, 15], [ 0, 1, 8, 9, 4, 5, 12, 13, 2, 3, 10, 11, 6, 7, 14, 15], [ 0, 8, 1, 9, 4, 12, 5, 13, 2, 10, 3, 11, 6, 14, 7, 15], [ 0, 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15], [ 0, 8, 4, 12, 1, 9, 5, 13, 2, 10, 6, 14, 3, 11, 7, 15], [ 0, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 15], [ 0, 4, 2, 6, 8, 12, 10, 14, 1, 5, 3, 7, 9, 13, 11, 15], [ 0, 2, 8, 10, 4, 6, 12, 14, 1, 3, 9, 11, 5, 7, 13, 15], [ 0, 8, 2, 10, 4, 12, 6, 14, 1, 9, 3, 11, 5, 13, 7, 15], [ 0, 4, 8, 12, 2, 6, 10, 14, 1, 5, 9, 13, 3, 7, 11, 15], [ 0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15]] |

## usage

These number patterns appear in many sources related to computer science.
Primer Preserving Permutations in music theory (PDF) :
| P0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| P0 | A | A | A | A | B | B | B | B | C | C | C | C | D | D | D | D |
| P1 | 0 | 1 | 4 | 5 | 2 | 3 | 6 | 7 | 8 | 9 | 12 | 13 | 10 | 11 | 14 | 15 |
| P1 | A | A | B | B | A | A | B | B | C | C | D | D | C | C | D | D |
| P2 | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 | 8 | 10 | 12 | 14 | 9 | 11 | 13 | 15 |
| P2 | A | B | A | B | A | B | A | B | C | D | C | D | C | D | C | D |
| P3 | 0 | 1 | 8 | 9 | 2 | 3 | 10 | 11 | 4 | 5 | 12 | 13 | 6 | 7 | 14 | 15 |
| P3 | A | A | B | B | C | C | D | D | A | A | B | B | C | C | D | D |
| P4 | 0 | 2 | 8 | 10 | 1 | 3 | 9 | 11 | 4 | 6 | 12 | 14 | 5 | 7 | 13 | 15 |
| P4 | A | B | A | B | C | D | C | D | A | B | A | B | C | D | C | D |
| P5 | 0 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 |
| P5 | A | B | C | D | A | B | C | D | A | B | C | D | A | B | C | D |

## signed

A signed Schoute permutation is a Schoute permutation permuted by the row of a XOR table. It corresponds to a signed permutation (but is not itself signed). This file shows all 16·24 = 384 signed Schoute permutations of degree 4. Each 16×16 matrix shows a Schoute permutation in the top row and its XOR permutations below.