

A319774


Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.


13




OFFSET

0,3


COMMENTS

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.


LINKS

Table of n, a(n) for n=0..5.


EXAMPLE

The a(3) = 14 set systems:
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1},{1,2},{1,3},{1,2,3}}
{{2},{1,2},{2,3},{1,2,3}}
{{3},{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}


MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#, Intersection[#1, #2]=={}&]&&stableQ[dual[#], Intersection[#1, #2]=={}&]&]], {n, 0, 3}] (* Gus Wiseman, Aug 19 2019 *)


CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting setsystems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting setsystems are A327039.
The BIInumbers of these setsystems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.
Sequence in context: A075044 A211891 A060599 * A065868 A144017 A032419
Adjacent sequences: A319771 A319772 A319773 * A319775 A319776 A319777


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Sep 27 2018


STATUS

approved



