Walsh permutation
The term nbit Walsh permutation was chosen by the author for permutations of 2^{n} elements, that permute Walsh functions into other Walsh functions.
Walsh permutations of 2^{n} elements can be represented by an nelement compression vector
of integers referring to the numbering of Walsh functions (displayed in orange in the images).
It corresponds to the n×n compression matrix. It's columns show the elements of the compression vector in binary representation.
The multiplication of the compression matrices (with F_{2} operations) corresponds to the composition of the permutations.
As the compression matrices are the elements of the general linear group GL(n,2) the Walsh permutations form a group isomorphic to GL(n,2).
There are A002884(n) nbit Walsh permutations.
Not all vectors with different elements correspond to Walsh permutations, as the following example shows:
No Walsh permutation wp(1,2,3,4)  


Notation warning[edit  edit source]
In this article and the related material it was chosen to base the compression vector on the columns of the compression matrix. But it would probably have been better to base it on the rows. Then the bit permutations and the small permutations corresponding to their compression vectors would be homomorphic. With the current convention they form an antihomomorphism. This might be corrected in the future.
With the row based convention the result of the example multiplication above would be wp(2, 3, 9, 15) instead of wp(14, 11, 8, 12).
Bit permutation[edit  edit source]
When a simple permutation of n elements is applied on the binary digits of numbers from 0 to 2^{n}1 the result is a permutation of the numbers from 0 to 2^{n}1.
The example on the right corresponds to the simple permutation that swaps the first two and the last two digits of the 4bit binary numbers.
Probably the most important bit permutation is the bitreversal permutation.
Nimber multiplication[edit  edit source]
The rows of each nimber multiplication table are Walsh permutations (except row 0).
They are not only closed under multiplication (function composition), but even under addition (bitwise XOR).
Gray code permutation powers[edit  edit source]
Powers of the Gray code permutation have very simple compression matrices and vectors.
In each vector all entries follow from the first one, and the first entries follow from the rows of the Sierpinski triangle.
Sequency ordered Walsh matrix[edit  edit source]
The permutation that changes the natural ordered into the sequency ordered Walsh matrix is the product of the Gray code permutation and the bitreversal permutation.
Bent functions[edit  edit source]
Each bent function corresponds to a Walsh permutation.
This can be seen when all rows of its sec matrix are XORed with the function itself (second step in the example on the right).
Magic squares[edit  edit source]
There are 24*9=216 Walsh permutations that correspond to magic squares of order 4.
One my say that only 6 of them are essentially different.
Inversions[edit  edit source]
Some inversion sets of Walsh permutations are very regular.
E.g. there are 2^{n}1 nbit Walsh permutations with horizontally striped inversion sets (like the left one of the examples).
The pattern of the stripes is that of Walsh functions.
2bit[edit  edit source]
The 2bit Walsh permutations form the general linear group GL(2,2) which is also the symmetric group S_{3}.
3bit[edit  edit source]
The A002884(3) = 168 3bit Walsh permutations correspond to the collineations of the Fano plane.
Compression vectors, Permutations
4bit[edit  edit source]
There are A002884(4) = 20160 4bit Walsh permutations.
Fixed points[edit  edit source]
The fixed points of an nbit Walsh permutation correspond to a subgroup of Z_{2}^{n}. (Compare Subgroups of nimber addition)