Permutations of Boolean functions

Analogous to hard and soft properties of Boolean functions, there are also hard and soft permutations.

A permutation is hard, when the domain is the infinite set of all Boolean functions.
An example is the map from a BF to its complement.

A permutation is soft, when the domain is the finite set of BF with a given arity.
An example is the map from a BF to its Zhegalkin twin.

Interesting permutations of BF are often Walsh permutations, which correspond to an invertible binary matrix of size $2^{arity}$.

Zhegalkin permutation

This might be the most important permutation of Boolean functions. It is soft.
A Boolean function can be represented by its truth table or its Zhegalkin index.
As a consequence, every other permutation of Boolean functions can be represented by four different permutation of integers:
Between truth tables, between Zhegalkin indices, and from one to the other.
Mathematically the one between Zhegalkin indices is probably most important. In the hard case, it is an infinite permutation.

Mentors of Boolean functions

The mentor permutation might be called semi-hard, because it is an infinite permutation between pairs of complements.

gradual sharpness

T to T:   (47803, 47800, 47549, 47537, 35996, 36012, 32960, 32768, 1, 769, 13617, 14641, 36253, 48541, 7517, 56669)
T to Z:   (65293, 43608, 52331, 34863, 61492, 41060, 49216, 32768, 65535, 43775, 52399, 34991, 61643, 41163, 49291, 32907)
Z to T:   (65535, 21845, 52428, 56797, 61680, 27756, 49344, 23901, 65280, 23280, 52224, 58476, 61440, 24768, 49152, 56669)
Z to Z:   (1, 3, 4, 11, 16, 36, 64, 139, 256, 528, 1024, 2084, 4096, 8256, 16384, 32907)   Template:Oeislink