Integer sequences related to Boolean functions

See also .../clans.

name	equivalent under	maximum size	sequence	0	1	2	3	4	5
family	neg	${\displaystyle 2^{n}}$	A000231	2	3	7	46	4336	134281216
splinter	intersection of family and faction		TBA7	2	4	14	184	51564
faction	perm	${\displaystyle n!}$	A003180 = 2 · A000612	2	4	12	80	3984	37333248
great faction	complementary and partner factions		A003180(n) / 4 for n > 0	1	1	3	20	996	9333312
clan	neg, perm	${\displaystyle 2^{n}\cdot n!}$	A000616	2	3	6	22	402	1228158
super-family blunt family	neg, comp	${\displaystyle 2^{n+1}}$	A000133	1	2	5	30	2288	67172352
super-clan blunt clan	neg, perm, comp	${\displaystyle 2^{n+1}\cdot n!}$	A000370	1	2	4	14	222	616126

A054724 number of families by weight
A039754 number of clans by weight

A227722 (families) and A227723 (clans) show smallest values corresponding to truth tables. Which Boolean function an integer represents, depends on the arity.

In the following the representative is always the smallest Zhegalkin index.

Balanced families and clans can contain their own complements.

The sequence for clans is a subsequence of the one for families. All entries are even.

A211344 ${\displaystyle T(n,n-k)}$ is the ${\displaystyle n}$-ary truth table of ${\displaystyle x_{k}}$

A227724 ${\displaystyle T(n,k)}$ is the number of balanced n-ary families of size ${\displaystyle 2^{k}}$
A227725 ${\displaystyle T(n,k)}$ is the number of ${\displaystyle n}$-ary families of size ${\displaystyle 2^{k}}$
A051502 ${\displaystyle a(n)}$ is the number of ${\displaystyle n}$-ary families of maximal size ${\displaystyle 2^{n}}$

A000371 ${\displaystyle a(n)}$ is the number of truly ${\displaystyle n}$-ary functions   (number of nondegenerate Boolean functions of ${\displaystyle n}$ variables)

Monotonic: A000372 (Dedekind numbers) counts functions, A003182 counts clans (which are probably the same as factions)