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A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system. 13
1, 1, 2, 14, 814, 1174774 (list; graph; refs; listen; history; text; internal format)
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 A144017 A032419

Adjacent sequences:  A319771 A319772 A319773 * A319775 A319776 A319777

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

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Last modified September 28 06:57 EDT 2021. Contains 347703 sequences. (Running on oeis4.)