A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

1, 1, 2, 14, 814, 1174774

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 set-systems are A051185.

The unlabeled multiset partition version is A319773.

The covering case is A327037.

The version without strict dual is A327038.

Cointersecting set-systems are A327039.

The BII-numbers of these set-systems are A327061.

Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

Sequence in context: A075044 A211891 A060599 * A065868 A032419 A144017

Adjacent sequences: A319771 A319772 A319773 * A319775 A319776 A319777

Keyword

nonn,more

Author

Gus Wiseman, Sep 27 2018

Status

approved