A326853 -id:A326853 - cp's OEIS frontend

A326786 Cut-connectivity of the set-system with BII-number n.

0, 1, 1, 0, 2, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Gus Wiseman, Jul 25 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (together with any resulting empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A326853), this is the same as vertex-connectivity (A327051).

			Positions of first appearances of each integer, together with the corresponding set-systems, are:
     0: {}
     1: {{1}}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

Cf. A000120, A013922, A029931, A048793, A070939, A305078, A322388, A322389 (same for MM-numbers), A322390, A326031, A326701, A326749, A326753, A326787 (edge-connectivity), A327051 (vertex-connectivity).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Min@@Length/@Select[Subsets[Union@@bpe/@y],Function[del,Length[csm[DeleteCases[DeleteCases[bpe/@y,Alternatives@@del,{2}],{}]]]!=1]]];
Table[vertConn[bpe[n]],{n,0,100}]

A327039 Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 2, 7, 88, 25421, 2077323118, 9221293242272922067, 170141182628636920942528022609657505092

Offset: 0

Gus Wiseman, Aug 17 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}}. This sequence counts set-systems that are cointersecting, meaning their dual is pairwise intersecting.

			The a(0) = 1 through a(2) = 7 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The unlabeled multiset partition version is A319752.
The BII-numbers of these set-systems are A326853.
The pairwise intersecting case is A327038.
The covering case is A327040.
The case where the dual is strict is A327052.
Cf. A058891, A306006, A319765, A327020, A327053.

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}]],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Binomial transform of A327040.

a(5)-a(7) from Christian Sievers, Oct 22 2023

A327040 Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 4, 72, 25104, 2077196832, 9221293229809363008, 170141182628636920877978969957369949312

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}}. This sequence counts covering set-systems that are cointersecting, meaning their dual is pairwise intersecting.

			The a(0) = 1 through a(2) = 4 set-systems:
  {}  {{1}}  {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}

The unlabeled multiset partition version is A319752.
The BII-numbers of these set-systems are A326853.
The antichain case is A327020.
The pairwise intersecting case is A327037.
The non-covering version is A327039.
The case where the dual is strict is A327053.
Cf. A003465, A305843, A305844, A306006, A319774, A327052.

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]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Inverse binomial transform of A327039.

a(5)-a(7) from Christian Sievers, Oct 22 2023

A327051 Vertex-connectivity of the set-system with BII-number n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Sep 02 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Except for cointersecting set-systems (A326853), this is the same as cut-connectivity (A326786).

			Positions of first appearances of each integer, together with the corresponding set-systems, are:
     0: {}
     4: {{1,2}}
    52: {{1,2},{1,3},{2,3}}
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

Wikipedia, k-vertex-connected graph

Cut-connectivity is A326786.
Spanning edge-connectivity is A327144.
Non-spanning edge-connectivity is A326787.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A029931, A048793, A070939, A259862, A322389, A323818, A326031, A327125, A327198, A327336.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]
Table[vertConnSys[Union@@bpe/@bpe[n],bpe/@bpe[n]],{n,0,100}]

A327098 BII-numbers of set-systems with cut-connectivity 1.

Original entry on oeis.org

1, 2, 8, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 128, 260, 261, 262, 263, 272, 273, 276, 277, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309

Offset: 1

Gus Wiseman, Aug 21 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   8: {{3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  36: {{1,2},{2,3}}
  37: {{1},{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  48: {{1,3},{2,3}}

A subset of A326749.
Positions of 1's in A326786.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 1 are A327111.
Integer partitions with cut-connectivity 1 are counted by A322390.
Labeled connected separable graphs are counted by A327114.
Connected separable set-systems are counted by A327197.
Cf. A000120, A048793, A070939, A322389, A326031, A327100, A327125.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==1&]

A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 17, 1451, 3741198

Offset: 0

Gus Wiseman, Aug 17 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}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains with union {1..n} whose dual is pairwise intersecting.

			The a(0) = 1 through a(4) = 17 antichains:
  {}  {{1}}  {{12}}  {{123}}         {{1234}}
                     {{12}{13}{23}}  {{12}{134}{234}}
                                     {{13}{124}{234}}
                                     {{14}{123}{234}}
                                     {{23}{124}{134}}
                                     {{24}{123}{134}}
                                     {{34}{123}{124}}
                                     {{123}{124}{134}}
                                     {{123}{124}{234}}
                                     {{123}{134}{234}}
                                     {{124}{134}{234}}
                                     {{12}{13}{14}{234}}
                                     {{12}{23}{24}{134}}
                                     {{13}{23}{34}{124}}
                                     {{14}{24}{34}{123}}
                                     {{123}{124}{134}{234}}
                                     {{12}{13}{14}{23}{24}{34}}

Covering, intersecting antichains are A305844.
Covering, T1 antichains are A319639.
Cointersecting set-systems are A327039.
Covering, cointersecting set-systems are A327040.
Covering, cointersecting set-systems are A327051.
The non-covering version is A327057.
Covering, intersecting, T1 set-systems are A327058.
Unlabeled cointersecting antichains of multisets are A327060.
Cf. A003465, A305843, A319765, A326853, A327037, A327038, A327053.

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}],SubsetQ],Union@@#==Range[n]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,4}]

Inverse binomial transform of A327057.

A327082 BII-numbers of set-systems with cut-connectivity 2.

Original entry on oeis.org

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 256, 257, 384, 385, 512, 514, 640, 642, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 20 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all set-systems with cut-connectivity 2 together with their BII-numbers begins:
    4: {{1,2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    7: {{1},{2},{1,2}}
   16: {{1,3}}
   17: {{1},{1,3}}
   24: {{3},{1,3}}
   25: {{1},{3},{1,3}}
   32: {{2,3}}
   34: {{2},{2,3}}
   40: {{3},{2,3}}
   42: {{2},{3},{2,3}}
  256: {{1,4}}
  257: {{1},{1,4}}
  384: {{4},{1,4}}
  385: {{1},{4},{1,4}}
  512: {{2,4}}
  514: {{2},{2,4}}
  640: {{4},{2,4}}
  642: {{2},{4},{2,4}}
The first term involving an edge of size 3 is 832: {{1,2,3},{1,4},{2,4}}.

Positions of 2's in A326786.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity 2 are A327108.
The cut-connectivity 1 version is A327098.
The cut-connectivity > 1 version is A327101.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
Cf. A000120, A002218, A013922, A048793, A259862, A322387, A322388, A326031, A327041, A327114.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Select[Range[0,100],vertConnSys[bpe/@bpe[#]]==2&]

A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

Original entry on oeis.org

4, 5, 6, 7, 16, 17, 24, 25, 32, 34, 40, 42, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Gus Wiseman, Aug 22 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

A set-system is 2-cut-connected if any single vertex can be removed (along with any empty edges) without making the set-system disconnected or empty. Except for cointersecting set-systems (A326853), this is the same as 2-vertex-connectivity.

			The sequence of all 2-cut-connected set-systems together with their BII-numbers begins:
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  53: {{1},{1,2},{1,3},{2,3}}
  54: {{2},{1,2},{1,3},{2,3}}
  55: {{1},{2},{1,2},{1,3},{2,3}}
  60: {{1,2},{3},{1,3},{2,3}}
  61: {{1},{1,2},{3},{1,3},{2,3}}
  62: {{2},{1,2},{3},{1,3},{2,3}}
  63: {{1},{2},{1,2},{3},{1,3},{2,3}}

Positions of numbers >= 2 in A326786.
2-cut-connected graphs are counted by A013922, if we assume A013922(2) = 0.
2-cut-connected integer partitions are counted by A322387.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
The labeled cut-connectivity triangle is A327125, with unlabeled version A327127.
Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Select[Range[0,100],cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

If (*) is intersection and (-) is complement, we have A327101 * A326704 = A326751 - A058891, i.e., the intersection of A327101 (this sequence) with A326704 (antichains) is the complement of A058891 (singletons) in A326751 (blobs).

A327057 Number of antichains covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 2, 4, 9, 36, 1572, 3750221

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}}. An antichain is a set of sets, none of which is a subset of any other. This sequence counts antichains whose dual is pairwise intersecting.

			The a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}       {}
      {{1}}  {{1}}    {{1}}
             {{2}}    {{2}}
             {{1,2}}  {{3}}
                      {{1,2}}
                      {{1,3}}
                      {{2,3}}
                      {{1,2,3}}
                      {{1,2},{1,3},{2,3}}

Antichains are A000372.
The BII-numbers of these set-systems are the intersection of A326704 and A326853.
The covering case is A327020.
Cointersecting set-systems are A327039.
Cf. A006126, A051185, A245567, A305844, A326950, A327038, A327052.

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}],SubsetQ],stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,5}]

Binomial transform of A327020.

A327062 Number of antichains of distinct sets covering a subset of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 5, 16, 81, 2595

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) = 16 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{2}}      {{2}}
             {{1,2}}    {{3}}
             {{1},{2}}  {{1,2}}
                        {{1,3}}
                        {{2,3}}
                        {{1},{2}}
                        {{1,2,3}}
                        {{1},{3}}
                        {{2},{3}}
                        {{1},{2,3}}
                        {{2},{1,3}}
                        {{3},{1,2}}
                        {{1},{2},{3}}
                        {{1,2},{1,3},{2,3}}

Antichains are A000372.
The covering case is A319639.
The non-isomorphic multiset partition version is A319721.
The BII-numbers of these set-systems are the intersection of A326910 and A326853.
Set-systems whose dual is a weak antichain are A326968.
The unlabeled version is A327018.
Cf. A006126, A318099, A293606, A326704, A326950, 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}],SubsetQ],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

cp's OEIS Frontend

A326786 Cut-connectivity of the set-system with BII-number n.

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A327039 Number of set-systems covering a subset of {1..n} where every two covered vertices appear together in some edge (cointersecting).

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A327040 Number of set-systems covering n vertices, every two of which appear together in some edge (cointersecting).

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A327051 Vertex-connectivity of the set-system with BII-number n.

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A327098 BII-numbers of set-systems with cut-connectivity 1.

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A327020 Number of antichains covering n vertices where every two vertices appear together in some edge (cointersecting).

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A327082 BII-numbers of set-systems with cut-connectivity 2.

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A327101 BII-numbers of 2-cut-connected set-systems (cut-connectivity >= 2).

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