Integer sequences related to Boolean functions

number of equivalence classes

See also .../clans.

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

by weight

Template:Oeislink number of families by weight
Template:Oeislink number of clans by weight

balanced

			1	2	3	4	5
families	balanced	Template:Color (central values of Template:Oeislink)	1	3	14	870	18796230
	self-complementary	Template:Oeislink	1	3	14	240	63488
clans	balanced	Template:Oeislink (central values of Template:Oeislink)	1	2	6	74	169112
	self-complementary	Template:Oeislink	1	2	6	42	4094

representatives of equivalence classes

Template:Oeislink (families) and Template:Oeislink (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.

self-complementary

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.

misc.

Template:Oeislink $T(n,n-k)$ is the $n$-ary truth table of $x_k$

Template:Oeislink $T(n,k)$ is the number of balanced n-ary families of size $2^k$
Template:Oeislink $T(n,k)$ is the number of $n$-ary families of size $2^k$
Template:Oeislink $a(n)$ is the number of $n$-ary families of maximal size $2^n$

Template:Oeislink $a(n)$ is the number of truly $n$-ary functions   (number of nondegenerate Boolean functions of $n$ variables)

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

		0	1	2	3	4
male	Template:Oeis	1	3	13	159	33337
female	Template:Oeis	1	1	3	97	32199

Template:Integer sequences related to Boolean functions/Google food