Legion and cohort of Boolean functions

Studies of Boolean functions

These properties are created from the truth table of a BF. The algorithm is illustrated in the images below.
Legion uses a top left Sierpinski triangle. Cohort uses the corresponding pattern of negated Walsh functions (see Walsh matrix).
True places in the truth table are dark columns in the matrix.
The result is the separate column on the right. It has a dot in every row without dotless dark matrix fields.

examples for 1001 0000

| 3-ary  | 4-ary  | result
legion  |  |  | {0, 4}
cohort  |  |  | {0, 3, 4, 7}

Every cohort belongs to a legion. Cohorts correspond to seals. (The cohort of a seal corresponds to its antipode.)

## trivial

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For most BF the result contains only the 0, which shall be called the trivial legion or cohort.

## special case contradiction

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For the contradiction the result is the set of all non-negative integers. (That is because the AND with no arguments is true. Compare empty product.)
That is not useful. The relationships with other properties suggest an adaptation:
The result for the contradiction shall be the same as that of the AND of all negated arguments (which is twin of the tautology).
With that changed definition these properties are soft, i.e. dependent on arity.

| hard  | soft

|
|

legion  | legion
gravity, legion weight, legion faction | soft legion
soft gravity, soft legion weight, soft legion faction
cohort  | cohort
cohort faction
depth, cohort weight | soft cohort
soft cohort faction
soft depth, soft cohort weight

Hereafter the terms legion and cohort are used for the soft properties.

## weight

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The weights of the resulting sets are always powers of two. Their exponents shall be called gravity and depth. (These are properties of seals, but have been generalized for all BF.)

The triangles Willow and ExOlive show the number of BF by arity and these properties.

## relationships

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Legion corresponds to twin atomvals, and gravity to twin valency.

There is a similar relationship for the cohort, but the key to it is not the twin. (So what is it?)

Cohort is related to symmetry neg, and depth to strength (the exponent of family size).

Two kinds of symmetry can be found in these matrices: Mirror symmetry along both diagonals, and 2-fold rotational symmetry.

symmetry  mirror  | rotational
Z (right)

|

T (left) | Z (right)
atomvals, valency | legion, gravity | cohort, depth
symmetry neg, strength

### integer partitions

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symmetry neg  arity  | partition

|

1  | 2⋅2
2  | 4⋅2 \+ 1⋅8
3  | 8⋅2 \+ 7⋅8 \+ 1⋅184
4  | 16⋅2 \+ 35⋅8 \+ 15⋅184 \+ 1⋅62464

The counts (to the left of the multiplication dot) form triangle Oak, but with the left two columns merged.
The sizes (to the right of the multiplication dot) form sequence ExAloe (with positive indices).