Template:Families of Boolean functions/sequences/arity to count

arity to count

			arity $n$
			0	1	2	3	4	5
	Saffron	balanced families	0	1	3	14	870	18796230
${\frac {(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}$	NonDubSage	haploid families	0	1	3	14	240	63488
${\frac {2^{2^{n}}}{2^{n}}}~=~2^{(2^{n}-n)}$	DubSage	diploid families (BF with a particular consul)	2	2	4	32	4096	134217728
$2^{n}-n-1$		Eulerian numbers	0	0	1	4	11	26
${\frac {2^{2^{n}}}{2^{n{\color {Red}+1}}}}~=~2^{(2^{n}-n-1)}$	Sage	sharp families (BF with a particular prefect)	1	1	2	16	2048	67108864
${\frac {2^{2^{n}}+(2^{2^{n-1}{\color {Red}+1}})(2^{n}-1)}{2^{n{\color {Red}+1}}}}$	NonSage	blunt families super-families	1	2	5	30	2288	67172352
${\frac {2^{2^{n}}+(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}$	Thyme	families	2	3	7	46	4336	134281216
	TornSage	sharp super-families		1	1	8	1024	33554432
	Clove	blunt super-families		1	4	22	1264	33617920
	Aloe	families with maximal size $2^{n}$ (diagonal of Olive)	2	1	2	23	3904	134156284
		blunt families with maximal size $2^{n}$	1	0	0	7	1856	67047420

formulas in Python

def f_haploid(n):
    a = 2 ** (2 ** (n - 1))
    b = 2 ** n - 1
    c = 2 ** n
    result = a * b / c
    assert result % 1 == 0  # is integer
    return int(result)


def f_diploid(n):
    a = 2 ** n - n
    result = 2 ** a
    assert result % 1 == 0  # is integer
    return int(result)


def f_sharp(n):
    a = 2 ** n - n - 1
    result = 2 ** a
    assert result % 1 == 0  # is integer
    return int(result)


def f_blunt(n):
    a = 2 ** (2 ** n)
    b = 2 ** (2 ** (n - 1) + 1)
    c = 2 ** n - 1
    d = 2 ** (n + 1)
    result = (a + b * c) / d
    assert result % 1 == 0  # is integer
    return int(result)


def f_total(n):
    a = 2 ** (2 ** n)
    b = 2 ** (2 ** (n - 1))
    c = 2 ** n - 1
    d = 2 ** n
    result = (a + b * c) / d
    assert result % 1 == 0  # is integer
    return int(result)


haploid = [0, 1, 3, 14, 240, 63488, 4227858432]
diploid = [2, 2, 4, 32, 4096, 134217728, 288230376151711744]
sharp = [1, 1, 2, 16, 2048, 67108864, 144115188075855872]
blunt = [1, 2, 5, 30, 2288, 67172352, 144115192303714304]
total = [2, 3, 7, 46, 4336, 134281216, 288230380379570176]


assert [f_haploid(n) for n in range(7)] == haploid
assert [f_diploid(n) for n in range(7)] == diploid
assert [f_sharp(n) for n in range(7)] == sharp
assert [f_blunt(n) for n in range(7)] == blunt
assert [f_total(n) for n in range(7)] == total


for n in range(7):

    assert haploid[n] == blunt[n] - sharp[n]
    assert diploid[n] == 2 * sharp[n]

    assert sharp[n] == blunt[n] - haploid[n]
    assert blunt[n] == sharp[n] + haploid[n] == (total[n] + haploid[n]) / 2

    assert total[n] == sharp[n] + blunt[n] == haploid[n] + diploid[n]

Families of Boolean functions#sequences

contains {{Families of Boolean functions/sequences/Python formulas}}