cp's OEIS Frontend

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

%I A319774 #13 Aug 18 2024 19:02:02
%S A319774 1,1,2,14,814,1174774,909125058112,291200434263385001951232
%N A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.
%C A319774 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}}.
%C A319774 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.
%e A319774 The a(3) = 14 set systems:
%e A319774    {{1},{1,2},{1,2,3}}
%e A319774    {{1},{1,3},{1,2,3}}
%e A319774    {{2},{1,2},{1,2,3}}
%e A319774    {{2},{2,3},{1,2,3}}
%e A319774    {{3},{1,3},{1,2,3}}
%e A319774    {{3},{2,3},{1,2,3}}
%e A319774    {{1,2},{1,3},{2,3}}
%e A319774    {{1,2},{1,3},{1,2,3}}
%e A319774    {{1,2},{2,3},{1,2,3}}
%e A319774    {{1,3},{2,3},{1,2,3}}
%e A319774    {{1},{1,2},{1,3},{1,2,3}}
%e A319774    {{2},{1,2},{2,3},{1,2,3}}
%e A319774    {{3},{1,3},{2,3},{1,2,3}}
%e A319774    {{1,2},{1,3},{2,3},{1,2,3}}
%t A319774 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A319774 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A319774 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 *)
%Y A319774 Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.
%Y A319774 Cf. A319752, A319765, A319766, A319767, A319768, A319769.
%Y A319774 Intersecting set-systems are A051185.
%Y A319774 The unlabeled multiset partition version is A319773.
%Y A319774 The covering case is A327037.
%Y A319774 The version without strict dual is A327038.
%Y A319774 Cointersecting set-systems are A327039.
%Y A319774 The BII-numbers of these set-systems are A327061.
%Y A319774 Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.
%K A319774 nonn,more
%O A319774 0,3
%A A319774 _Gus Wiseman_, Sep 27 2018
%E A319774 a(6)-a(7) from _Christian Sievers_, Aug 18 2024