Schoute matrix

Schoute matrix rows: Schoute permutations columns: Schoute partitions

The Schoute matrices are two infinite matrices, whose rows are periodic permutations of the non-negative integers.
These Schoute permutations are the result of applying finite permutations to the binary digits of the non-negative integers.
The order of the rows follows from the reverse colexicographic order of finite permutations.

The columns can be seen as partitions of the set of finite permutations, which shall be called Schoute partitions.

There appear to be no common names for these concepts. So here they have been named after Pieter Schoute, who probably first described the permutohedron.

The two versions shall be called passive and active.
The passive one shows the results of applying the respective bit-permutation. It is described by  A195665.
The active one shows the permutations that produce these results. (A permutation is a bijective map. The result of applying it looks like its inverse.)

passive

active

factions of Boolean functions

These are three related factions of 4-ary Boolean functions. (See also Smallest Zhegalkin index#Factions.)
These binary matrices can be seen as passive Schoute matrices, where each integer is assigned a Boolean value.

two twin factions ...

... and the corresponding noble faction