Template:Gender of Boolean functions/triangles

total

This triangle shows the numbers of Boolean functions with arity $n$ and valency $k$ .
The right diagonal is ${\boldsymbol {p}}$ = A000371. ${\boldsymbol {p}}(n)$ is the number of non-degenerate $n$ -ary Boolean functions. (See root.)
This is Pascal's triangle with column $k$ multiplied by ${\boldsymbol {p}}(k)$ .

The sum of row $n$ is $2^{2^{n}}$ . ( A001146)

k n	0	1	2	3	4	5	sums
0	1 · 2 2						2
1	1 · 2 2	1 · 2 2					4
2	1 · 2 2	2 · 2 4	1 · 10 10				16
3	1 · 2 2	3 · 2 6	3 · 10 30	1 · 218 218			256
4	1 · 2 2	4 · 2 8	6 · 10 60	4 · 218 872	1 · 64594 64594		65536
5	1 · 2 2	5 · 2 10	10 · 10 100	10 · 218 2180	5 · 64594 322970	1 · 4294642034 4294642034	4294967296

♀

This triangle shows the numbers of female Boolean functions with arity $n$ and valency $k$ .
The right diagonal is ${\boldsymbol {d}}={\boldsymbol {p}}-{\boldsymbol {q}}$ = 1, 0, 2, 90, 31826, 2147158386... (compare A007537)
This is Pascal's triangle with column $k$ multiplied by ${\boldsymbol {d}}(k)$ .

The row sums are A246537.

k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 1	1 · 0 0					1
2	1 · 1 1	2 · 0 0	1 · 2 2				3
3	1 · 1 1	3 · 0 0	3 · 2 6	1 · 90 90			97
4	1 · 1 1	4 · 0 0	6 · 2 12	4 · 90 360	1 · 31826 31826		32199
5	1 · 1 1	5 · 0 0	10 · 2 20	10 · 90 900	5 · 31826 159130	1 · 2147158386 2147158386	2147318437

♂

This triangle shows the numbers of male Boolean functions with arity $n$ and valency $k$ .
The right diagonal is ${\boldsymbol {q}}(n)=2^{(2^{n}-1)}$ . (That is half the number of all $n$ -ary Boolean functions.) (like A058891, but with offset 0)
This is Pascal's triangle with column $k$ multiplied by ${\boldsymbol {q}}(k)$ .

The row sums are A246418.

k n	0	1	2	3	4	5	sums
0	1 · 1 1						1
1	1 · 1 1	1 · 2 2					3
2	1 · 1 1	2 · 2 4	1 · 8 8				13
3	1 · 1 1	3 · 2 6	3 · 8 24	1 · 128 128			159
4	1 · 1 1	4 · 2 8	6 · 8 48	4 · 128 512	1 · 32768 32768		33337
5	1 · 1 1	5 · 2 10	10 · 8 80	10 · 128 1280	5 · 32768 163840	1 · 2147483648 2147483648	2147648859

from math import factorial


def binom(n, k):
    return factorial(n) // (factorial(k) * factorial(n - k))


for n in range(6):
    row = []
    for k in range(n+1):
        row.append(
            binom(n, k) * 2 ** (2 ** k - 1)
        )
    print(n, '-->', row, '-->', sum(row))

illustrated triangles

These images show the male functions also seen in the two 8×256 matrices. (Only a few of the 128 on the right side of the matrix are shown.)
They are ordered by adicity (vertical) and valency (horizontal).
The numbers sum up to those in the triangle above. (E.g. 2 + 2 + 2 = 6 and 8 + 16 = 24.)

Template:Gender of Boolean functions/triangles

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