A327334 -id:A327334 - cp's OEIS frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 11, 7, 2, 1, 44, 56, 39, 13, 3, 1, 191, 385, 332, 111, 21, 3, 1, 1229, 3994, 4735, 2004, 345, 34, 4, 1, 13588, 67014, 113176, 66410, 13429, 992, 54, 4, 1, 288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1, 12297299, 105731474, 327695586, 388624106, 162318088, 21500415, 820956, 9813, 121, 5, 1

Offset: 1

N. J. A. Sloane, Jul 08 2015

These are vertex-connectivities. Spanning edge-connectivity is A263296. Non-spanning edge-connectivity is A327236. Cut-connectivity is A327127. - Gus Wiseman, Sep 03 2019

			Triangle begins:
       1;
       1,       1;
       2,       1,       1;
       5,       3,       2,       1;
      13,      11,       7,       2,       1;
      44,      56,      39,      13,       3,     1;
     191,     385,     332,     111,      21,     3,    1;
    1229,    3994,    4735,    2004,     345,    34,    4,  1;
   13588,   67014,  113176,   66410,   13429,   992,   54,  4, 1;
  288597, 1973029, 4629463, 3902344, 1109105, 99419, 3124, 81, 5, 1;
  12297299,105731474,327695586,388624106,162318088,21500415,820956,9813,121,5,1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78
Brendan McKay, confusion over k-connected graphs, posting to Sequence Fans Mailing List, Jul 08 2015.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, The graphs counted by row n = 5 (isolated vertices not shown).

Columns k=0..10 (up to initial nonzero terms) are A000719, A052442, A052443, A052444, A052445, A324234, A324235, A324088, A324089, A324090, A324091.
Row sums are A000088.
Number of graphs with connectivity at least k for k=1..10 are A001349, A002218, A006290, A086216, A086217, A324240, A324092, A324093, A324094, A324095.
The labeled version is A327334.
Cf. A002494, A263296, A322389, A326786, A327127, A327236.

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&]

A052443 Number of simple unlabeled n-node graphs of connectivity 2.

Original entry on oeis.org

0, 0, 1, 2, 7, 39, 332, 4735, 113176, 4629463, 327695586, 40525166511, 8850388574939, 3453378695335727, 2435485662537561705, 3137225298932374490227, 7448146273273417700880931, 32837456713651735794742705141, 270528237651574516777595556494978, 4186091025846007046878947026003803389

Offset: 1

Eric W. Weisstein

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..23
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Connected Graph.
Gus Wiseman, The a(5) = 7 graphs with vertex-connectivity 2.

Column k=2 of A259862.
Cf. A000719, A002218, A006290, A052442, A052444, A052445.
The labeled version is A327198.
2-vertex-connected graphs are A013922.
Cf. A322387, A327051, A327101, A327334.

A002218 = Cases[Import["https://oeis.org/A002218/b002218.txt", "Table"], {, }][[All, 2]];
A006290 = Cases[Import["https://oeis.org/A006290/b006290.txt", "Table"], {, }][[All, 2]];
a[1] = 0; a[2] = 0; a[3] = 1;
a[n_] := A002218[[n]] - A006290[[n-3]];
Array[a, 23] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

a(n) = A002218(n) - A006290(n) for n > 2. - Andrew Howroyd, Sep 04 2019

Name clarified and a(8)-a(11) by Jens M. Schmidt, Feb 18 2019
a(2)-a(3) corrected by Andrew Howroyd, Aug 28 2019
a(12)-a(20) from Andrew Howroyd, Sep 04 2019

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&]

A327336 Number of labeled simple graphs with vertex-connectivity 1.

Original entry on oeis.org

0, 0, 1, 3, 28, 490, 15336, 851368, 85010976, 15615858960, 5388679220480, 3548130389657216, 4507988483733389568, 11145255551131555572992, 53964198507018134569758720, 514158235191699333805861463040, 9672967865350359173180572164444160

Offset: 0

Gus Wiseman, Sep 02 2019

nonn

Same as A327114 except a(2) = 1.

The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton.

			The a(2) = 1 through a(4) = 28 edge-sets:
  {12}  {12,13}  {12,13,14}
        {12,23}  {12,13,24}
        {13,23}  {12,13,34}
                 {12,14,23}
                 {12,14,34}
                 {12,23,24}
                 {12,23,34}
                 {12,24,34}
                 {13,14,23}
                 {13,14,24}
                 {13,23,24}
                 {13,23,34}
                 {13,24,34}
                 {14,23,24}
                 {14,23,34}
                 {14,24,34}
                 {12,13,14,23}
                 {12,13,14,24}
                 {12,13,14,34}
                 {12,13,23,24}
                 {12,13,23,34}
                 {12,14,23,24}
                 {12,14,24,34}
                 {12,23,24,34}
                 {13,14,23,34}
                 {13,14,24,34}
                 {13,23,24,34}
                 {14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327334.
The unlabeled version is A052442.
Connected non-separable graphs are A013922.
Set-systems with vertex-connectivity 1 are A327128.
Labeled simple graphs with cut-connectivity 1 are A327114.
Cf. A006129, A054592, A322389, A322390, A326786, A327070, A327098, A327100, A327125, A327126.

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[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==1&]],{n,0,4}]

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 3, 2, 0, 30, 40, 27, 17, 0, 546, 1365, 1842, 1690, 1451, 0, 41334

Offset: 0

Gus Wiseman, Sep 09 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
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. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.

			Triangle begins:
    1
    1    0
    1    1    0
    4    3    2    0
   30   40   27   17    0
  546 1365 1842 1690 1451    0

Row sums are A307249, or A006126 if empty edges are allowed.
Column k = 0 is A120338, if we assume A120338(0) = A120338(1) = 1.
Column k = 1 is A327356.
Column k = n - 1 is A327020.
The unlabeled version is A327359.
The version for vertex-connectivity >= k is A327350.
The version for spanning edge-connectivity is A327352.
The version for non-spanning edge-connectivity is A327353, with covering case A327357.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336.

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]]]]]]]]];
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]]]];
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[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

a(21) from Robert Price, May 28 2021

A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

Original entry on oeis.org

0, 0, 4, 72, 29856

Offset: 0

Gus Wiseman, Aug 24 2019

A set-system is a finite set of finite nonempty sets. 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 (along with any empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).

			Non-isomorphic representatives of the a(3) = 72 set-systems:
  {{123}}
  {{3}{123}}
  {{23}{123}}
  {{2}{3}{123}}
  {{1}{23}{123}}
  {{3}{23}{123}}
  {{12}{13}{23}}
  {{13}{23}{123}}
  {{1}{2}{3}{123}}
  {{1}{3}{23}{123}}
  {{2}{3}{23}{123}}
  {{3}{12}{13}{23}}
  {{2}{13}{23}{123}}
  {{3}{13}{23}{123}}
  {{12}{13}{23}{123}}
  {{1}{2}{3}{23}{123}}
  {{2}{3}{12}{13}{23}}
  {{1}{2}{13}{23}{123}}
  {{2}{3}{13}{23}{123}}
  {{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}}
  {{1}{2}{3}{13}{23}{123}}
  {{2}{3}{12}{13}{23}{123}}
  {{1}{2}{3}{12}{13}{23}{123}}

Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1.
Covering 1-cut-connected antichains (clutters) are A048143, if we assume A048143(0) = A048143(1) =0.
Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0.
Covering set-systems with cut-connectivity 2 are A327113.
2-vertex-connected integer partitions are A322387.
BII-numbers of set-systems with cut-connectivity >= 2 are A327101.
The cut-connectivity of the set-system with BII-number n is A326786(n).
Cf. A002218, A003465, A013922, A259862, A323818, A327082, A327126, A327130.

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]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]>=2&]],{n,0,3}]

A327113 Number of set-systems covering n vertices with cut-connectivity 2.

Original entry on oeis.org

0, 0, 4, 0, 4752

Offset: 0

Gus Wiseman, Aug 24 2019

A set-system is a finite set of finite nonempty sets. 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 (along with any empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).

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

Covering graphs with cut-connectivity >= 2 are A013922, if we assume A013922(2) = 1.
Covering antichains (blobs) with cut-connectivity >= 2 are A275307, if we assume A275307(1) = 0.
2-vertex-connected integer partitions are A322387.
Connected covering set-systems are A323818.
Covering set-systems with cut-connectivity >= 2 are A327112.
The cut-connectivity of the set-system with BII-number n is A326786(n).
BII-numbers of set-systems with cut-connectivity 2 are A327082.
Cf. A002218, A003465, A048143, A259862, A322389, A327101, A327113, A327126, A327128, A327130.

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]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]==2&]],{n,0,3}]

A327350 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 9, 5, 2, 0, 114, 84, 44, 17, 0, 6894, 6348, 4983, 3141, 1451, 0, 7785062

Offset: 0

Gus Wiseman, Sep 09 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
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. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.

			Triangle begins:
     1
     1    0
     2    1    0
     9    5    2    0
   114   84   44   17    0
  6894 6348 4983 3141 1451    0
The antichains counted in row n = 3:
  {123}         {123}         {123}
  {1}{23}       {12}{13}      {12}{13}{23}
  {2}{13}       {12}{23}
  {3}{12}       {13}{23}
  {12}{13}      {12}{13}{23}
  {12}{23}
  {13}{23}
  {1}{2}{3}
  {12}{13}{23}

Column k = 0 is A307249, or A006126 if empty edges are allowed.
Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.
Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.
Column k = n - 1 is A327020 (cointersecting antichains).
The unlabeled version is A327358.
Negated first differences of rows are A327351.
BII-numbers of antichains are A326704.
Antichain covers are A006126.
Cf. A003465, A014466, A120338, A293606, A293993, A319639, A323818, A327112, A327125, A327334, A327336, A327352, A327356, A327357, A327358.

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]]]]]]]]];
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]]]];
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[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]

a(21) from Robert Price, May 24 2021

cp's OEIS Frontend

A259862 Triangle read by rows: T(n,k) = number of unlabeled graphs with n nodes and connectivity exactly k (n>=1, 0<=k<=n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A052443 Number of simple unlabeled n-node graphs of connectivity 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327336 Number of labeled simple graphs with vertex-connectivity 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A327113 Number of set-systems covering n vertices with cut-connectivity 2.

Views

Author