A327059 -id:A327059 - cp's OEIS frontend

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

1, 1, 1, 3, 155

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(3) = 3 set-systems:
  {}  {{1}}  {{12}}  {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Covering intersecting set-systems are A305843.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The non-covering version is A327059.
The unlabeled multiset partition version is A327060.
Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A327059.

A327060 Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 3, 4, 9, 11, 30, 42, 103, 194, 443

Offset: 0

Gus Wiseman, Aug 18 2019

A multiset partition is a finite multiset of finite nonempty multisets. It is a weak antichain if no part is a proper submultiset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 multiset partitions:
  {}  {{1}}  {{11}}    {{111}}      {{1111}}        {{11111}}
             {{12}}    {{122}}      {{1122}}        {{11222}}
             {{1}{1}}  {{123}}      {{1222}}        {{12222}}
                       {{1}{1}{1}}  {{1233}}        {{12233}}
                                    {{1234}}        {{12333}}
                                    {{11}{11}}      {{12344}}
                                    {{12}{12}}      {{12345}}
                                    {{12}{22}}      {{11}{122}}
                                    {{1}{1}{1}{1}}  {{12}{222}}
                                                    {{33}{123}}
                                                    {{1}{1}{1}{1}{1}}

Antichains are A000372.
The BII-numbers of these set-systems are the intersection of A326853 and A326704.
Cointersecting set-systems are A327039.
The set-system version is A327057, with covering case A327058.
Cf. A006126, A007716, A051185, A305844, A326965, A327020, A327059, A327062.

cp's OEIS Frontend

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

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