View source for Integer sequences related to Boolean functions

{{Boolf header}}
__NOTOC__

==number of equivalence classes==

See also [[Integer sequences related to Boolean functions/clans|.../clans]].

{| class="wikitable" style="text-align: center;"
! name
! equivalent under
! maximum size
! sequence
! 0 !! 1 !! 2 !! 3 !! 4 !! 5
|-
| '''family'''
| neg
| <math>2^n</math>
| {{oeislink|A000231}}
| 2 || 3 || 7 || 46 || 4336 || 134281216
|- style="background-color: #fefbf7;"
| splinter
|colspan="2"| <small style="opacity: .5;">intersection of family and faction</small>
| {{color|red|TBA7}}
| 2 || 4 || 14 || 184 || 51564 || 
|- style="background-color: #fefbf7;"
| '''faction'''
| perm
| <math>n!</math>
| {{oeislink|A003180}} <small>= 2 &middot; {{oeislink|A000612}}</small>
| 2 || 4 || 12 || 80 || 3984 || 37333248
|- style="background-color: #fefbf7;"
| great faction
|colspan="2"| <small style="opacity: .5;">complementary and partner factions</small>
| <small>{{oeislink|A003180}}(n) / 4 for n > 0</small>
| <span style="opacity: .4;">1</span> || 1 || 3 || 20 || 996 || 9333312
|-
| '''clan'''
| neg,  perm
| <math>2^n \cdot n!</math>
| {{oeislink|A000616}}
| 2 || 3 || 6 || 22 || 402 || 1228158
|-
| super-family<br>blunt family
| neg, comp
| <math>2^{n+1}</math>
| {{oeislink|A000133}}
| 1 || 2 || 5 || 30 || 2288 || 67172352
|-
|-
| super-clan<br>blunt clan
| neg, perm, comp
| <math>2^{n+1} \cdot n!</math>
| {{oeislink|A000370}}
| 1 || 2 || 4 || 14 || 222 || 616126
|}

==by weight==

{{oeislink|A054724}} number of families by weight<br>
{{oeislink|A039754}} number of clans by weight

===balanced===

{| class="wikitable" style="text-align: center;"
!colspan="3"|  !! 1 !! 2 !! 3 !! 4 !! 5
|-
!rowspan="2"| families
! balanced
!style="font-weight: normal;"| {{color|red|TBA1}} <small>(central values of {{oeislink|A054724}})</small>
| 1 || 3 || 14 || 870 || 18796230
|-
! self-complementary
!style="font-weight: normal;"| {{oeislink|A001320}}
| 1 || 3 || 14 || 240 || 63488
|- style="border-top: 2px solid #a2a9b1;"
!rowspan="2"| clans
! balanced
!style="font-weight: normal;"| {{oeislink|A000721}} <small>(central values of {{oeislink|A039754}})</small>
| 1 || 2 || 6 || 74 || 169112
|-
! self-complementary
!style="font-weight: normal;"| {{oeislink|A000610}}
| 1 || 2 || 6 || 42 || 4094
|}

==representatives of equivalence classes==

<small>
{{oeislink|A227722}} (families) and {{oeislink|A227723}} (clans) show smallest values corresponding to truth tables.
Which Boolean function an integer represents, depends on the arity.
</small>

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

{{Integer sequences related to Boolean functions/reps of families}}
{{Integer sequences related to Boolean functions/reps of splinters}}
{{Integer sequences related to Boolean functions/reps of factions}}
{{Integer sequences related to Boolean functions/reps of clans}}

===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.<br>

{{Integer sequences related to Boolean functions/reps of self-comp families}}

{{Integer sequences related to Boolean functions/reps of self-comp clans}}

==misc.==

{{oeislink|A211344}} <math>T(n,n-k)</math> is the <math>n</math>-ary truth table of <math>x_k</math>

{{oeislink|A227724}} <math>T(n,k)</math> is the number of balanced ''n''-ary families of size <math>2^k</math><br>
{{oeislink|A227725}} <math>T(n,k)</math> is the number of <math>n</math>-ary families of size <math>2^k</math><br>
{{oeislink|A051502}} <math>a(n)</math> is the number of <math>n</math>-ary families of maximal size <math>2^n</math><br>

{{oeislink|A000371}} <math>a(n)</math> is the number of truly <math>n</math>-ary functions &nbsp; <small>(number of nondegenerate Boolean functions of <math>n</math> variables)</small>

Monotonic: {{oeislink|A000372}} (Dedekind numbers) counts functions, {{oeislink|A003182}} counts clans <small>(which are probably the same as factions)
</small>
{| class="wikitable" style="text-align: center;"
!  !!  !! 0 !! 1 !! 2 !! 3 !! 4
|-
! [[Boolf-term#male|male]]
! {{oeis|A246418}}
| 1 || 3 || 13 || 159 || 33337
|-
! [[Boolf-term#female|female]]
! {{oeis|A246537}}
| 1 || 1 || 3 || 97 || 32199
|}


{{Integer sequences related to Boolean functions/Google food}}


[[Category:Boolf sequences/old stuff]]