A327334 -id:A327334 - cp's OEIS frontend

A327100 BII-numbers of antichains of sets with cut-connectivity 1.

1, 2, 8, 20, 36, 48, 128, 260, 272, 276, 292, 304, 308, 320, 516, 532, 544, 548, 560, 564, 576, 768, 784, 788, 800, 804, 1040, 1056, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304, 2308, 2324, 2336, 2352, 2560, 2564, 2576, 2596, 2608, 2816, 2820, 2832, 2848

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.

We define the cut-connectivity 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 has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

			The sequence of all antichains of sets with vertex-connectivity 1 together with their BII-numbers begins:
    1: {{1}}
    2: {{2}}
    8: {{3}}
   20: {{1,2},{1,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
  128: {{4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}
  516: {{1,2},{2,4}}
  532: {{1,2},{1,3},{2,4}}
  544: {{2,3},{2,4}}
  548: {{1,2},{2,3},{2,4}}
  560: {{1,3},{2,3},{2,4}}
  564: {{1,2},{1,3},{2,3},{2,4}}

Positions of 1's in A326786.
The graphical case is A327114.
BII numbers of antichains with vertex-connectivity >= 1 are A326750.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
Cf. A000120, A000372, A006126, A048143, A048793, A070939, A322390, A326031, A326749, A326751, A327071, A327111.

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]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
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],stableQ[bpe/@bpe[#],SubsetQ]&&cutConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

If (+) is union and (-) is complement, we have A327100 = A058891 + (A326750 - A326751).

A327358 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 5, 3, 2, 0, 20, 14, 10, 6, 0, 180, 157, 128, 91, 54, 0

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
    5   3   2   0
   20  14  10   6   0
  180 157 128  91  54   0
Non-isomorphic representatives of the antichains counted in row n = 4:
  {1234}          {1234}           {1234}           {1234}
  {1}{234}        {12}{134}        {123}{124}       {12}{134}{234}
  {12}{34}        {123}{124}       {12}{13}{234}    {123}{124}{134}
  {12}{134}       {12}{13}{14}     {12}{134}{234}   {12}{13}{14}{234}
  {123}{124}      {12}{13}{24}     {123}{124}{134}  {123}{124}{134}{234}
  {1}{2}{34}      {12}{13}{234}    {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
  {2}{13}{14}     {12}{134}{234}   {12}{13}{14}{234}
  {12}{13}{14}    {123}{124}{134}  {12}{13}{14}{23}{24}
  {12}{13}{24}    {12}{13}{14}{23} {123}{124}{134}{234}
  {1}{2}{3}{4}    {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
  {12}{13}{234}   {12}{13}{14}{234}
  {12}{134}{234}  {12}{13}{14}{23}{24}
  {123}{124}{134} {123}{124}{134}{234}
  {4}{12}{13}{23} {12}{13}{14}{23}{24}{34}
  {12}{13}{14}{23}
  {12}{13}{24}{34}
  {12}{13}{14}{234}
  {12}{13}{14}{23}{24}
  {123}{124}{134}{234}
  {12}{13}{14}{23}{24}{34}

Column k = 0 is A261005, or A006602 if empty edges are allowed.
Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
Column k = n - 1 is A327425 (cointersecting).
The labeled version is A327350.
Negated first differences of rows are A327359.
Cf. A006126, A055621, A120338, A293606, A293993, A327334, A327351, A327356.

A327366 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and minimum vertex-degree k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 3, 1, 0, 23, 31, 9, 1, 0, 256, 515, 227, 25, 1, 0, 5319, 15381, 10210, 1782, 75, 1, 0, 209868, 834491, 815867, 221130, 15564, 231, 1, 0, 15912975, 83016613, 116035801, 47818683, 5499165, 151455, 763, 1, 0, 2343052576, 15330074139, 29550173053, 18044889597, 3291232419, 158416629, 1635703, 2619, 1, 0

Offset: 0

Gus Wiseman, Sep 04 2019

The minimum vertex-degree of the empty graph is infinity. It has been included here under k = 0. - Andrew Howroyd, Mar 09 2020

			Triangle begins:
     1
     1     0
     1     1     0
     4     3     1     0
    23    31     9     1     0
   256   515   227    25     1     0
  5319 15381 10210  1782    75     1     0

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows n = 0..20)
Gus Wiseman, The graphs with 4 vertices and minimum vertex-degree k (row n = 4).

Row sums are A006125.
Row sums without the first column are A006129.
Row sums without the first two columns are A100743.
Column k = 0 is A327367(n > 0).
Column k = 1 is A327227.
The unlabeled version is A294217.
Cf. A059167, A245797, A327069, A327103, A327334, A327369.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],k==If[#=={}||Union@@#!=Range[n],0,Min@@Length/@Split[Sort[Join@@#]]]&]],{n,0,5},{k,0,n}]

GraphsByMaxDegree(n)={
  local(M=Map(Mat([x^0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(merge(r, p, v)=acc(p + sum(i=1, poldegree(p)-r-1, polcoef(p, i)*(1-x^i)), v));
  my(recurse(r, p, i, q, v, e)=if(i<0, merge(r, x^e+q, v), my(t=polcoef(p, i)); for(k=0, t, self()(r, p, i-1, (t-k+x*k)*x^i+q, binomial(t, k)*v, e+k))));
  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0)));
  Mat(M);
}
Row(n)={if(n==0, [1], my(M=GraphsByMaxDegree(n), u=vector(n+1)); for(i=1, matsize(M)[1], u[n-poldegree(M[i,1])]+=M[i,2]); u)}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 09 2020

Terms a(28) and beyond from Andrew Howroyd, Sep 09 2019

A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 0, 6, 4, 4, 6, 0, 23, 29, 37, 37, 54, 0

Offset: 0

Gus Wiseman, Sep 10 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
   2  1  2  0
   6  4  4  6  0
  23 29 37 37 54  0
Row n = 4 counts the following antichains:
{1}{234}      {14}{234}        {134}{234}           {1234}
{12}{34}      {13}{24}{34}     {13}{14}{234}        {12}{134}{234}
{1}{2}{34}    {14}{24}{34}     {12}{13}{24}{34}     {124}{134}{234}
{1}{24}{34}   {14}{23}{24}{34} {13}{14}{23}{24}{34} {12}{13}{14}{234}
{1}{2}{3}{4}                                        {123}{124}{134}{234}
{1}{23}{24}{34}                                     {12}{13}{14}{23}{24}{34}

Row sums are A261005, or A006602 if empty edges are allowed.
Column k = 0 is A327426.
Column k = 1 is A327436.
Column k = n - 1 is A327425.
The labeled version is A327351.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336, A327350, A327358.

A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

Original entry on oeis.org

0, 1, 2, 27, 2084

Offset: 0

Gus Wiseman, Sep 02 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. We define the cut-connectivity (A326786, A327237, A327126) 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 has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).

The covering version is A327197.
The BII-numbers of these set-systems are A327098.
Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327113, A327114, A327126, A327229.

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]]]]]]]]];
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]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],cutConnSys[Union@@#,#]==1&]],{n,0,3}]

Binomial transform of A327197.

A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

Original entry on oeis.org

0, 0, 1, 3, 40, 1365

Offset: 0

Gus Wiseman, Sep 11 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 resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Non-isomorphic representatives of the a(4) = 40 set-systems:
  {{1,2},{1,3,4}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}

Column k = 1 of A327351.
The graphical case is A327336.
The unlabeled version is A327436.
Cf. A014466, A048143, A307249, A326786, A326950, A327062, A327112, A327114, A327334.

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],vertConnSys[Range[n],#]==1&]],{n,0,4}]

A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 2, 1, 0, 11, 6, 3, 1, 0, 34, 21, 10, 3, 1, 0, 156, 112, 56, 17, 4, 1, 0, 1044, 853, 468, 136, 25, 4, 1, 0, 12346, 11117, 7123, 2388, 384, 39, 5, 1, 0, 274668, 261080, 194066, 80890, 14480, 1051, 59, 5, 1, 0, 12005168, 11716571, 9743542, 5114079, 1211735, 102630, 3211, 87, 6, 1, 0

Offset: 0

Gus Wiseman, Sep 26 2019

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. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
   1
   1  0
   2  1  0
   4  2  1  0
  11  6  3  1  0
  34 21 10  3  1  0

Gus Wiseman, The graphs counted in row n = 4 (isolated vertices not shown).

Row-wise partial sums of A259862.
The labeled version is A327363.
The covering case is A327365, from which this sequence differs only in the k = 0 column.
Column k = 0 is A000088 (graphs).
Column k = 1 is A001349 (connected graphs), if we assume A001349(0) = A001349(1) = 0.
Column k = 2 is A002218 (2-connected graphs), if we assume A002218(2) = 0.
The triangle for vertex-connectivity exactly k is A259862.
Cf. A326786, A327051, A327114, A327125, A327126, A327127, A327334.

T(n,k) = Sum_{j=k..n} A259862(n,j).

Terms a(21) and beyond from Andrew Howroyd, Dec 26 2020

A327374 BII-numbers of set-systems with vertex-connectivity 2.

Original entry on oeis.org

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, 108, 109, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Gus Wiseman, Sep 04 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 resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			The sequence of all set-systems with vertex-connectivity 2 together with their BII-numbers begins:
  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}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  67: {{1},{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}
  73: {{1},{3},{1,2,3}}
  74: {{2},{3},{1,2,3}}
  75: {{1},{2},{3},{1,2,3}}

Positions of 2's in A327051.
Cut-connectivity 2 is A327082.
Spanning edge-connectivity 2 is A327108.
Non-spanning edge-connectivity 2 is A327097.
Vertex-connectivity 3 is A327376.
Labeled graphs with vertex-connectivity 2 are A327198.
Set-systems with vertex-connectivity 2 are A327375.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A048793, A070939, A259862, A326031, A326749, 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]}]];
Select[Range[0,200],vertConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

A327375 Number of set-systems with n vertices and vertex-connectivity 2.

Original entry on oeis.org

0, 0, 0, 72, 4752

Offset: 0

Gus Wiseman, Sep 05 2019

A set-system is a finite set of finite nonempty sets. 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 resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

BII-numbers for vertex-connectivity 2 are A327374.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
Labeled graphs with vertex-connectivity 2 are A327198.
The vertex-connectivity of the set-system with BII-number n is A327051(n).
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A013922, A259862, A322389, A322390, A323818, 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]]]]]]]]];
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],{1,n}]],vertConnSys[Range[n],#]==2&]],{n,0,3}]

A327436 Number of connected, unlabeled antichains of nonempty subsets of {1..n} covering n vertices with at least one cut-vertex (vertex-connectivity 1).

Original entry on oeis.org

0, 0, 1, 1, 4, 29

Offset: 0

Gus Wiseman, Sep 11 2019

nonn

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 29 antichains:
  {12}  {12}{13}  {12}{134}         {12}{1345}
                  {12}{13}{14}      {123}{145}
                  {12}{13}{24}      {12}{13}{145}
                  {12}{13}{14}{23}  {12}{13}{245}
                                    {13}{24}{125}
                                    {13}{124}{125}
                                    {14}{123}{235}
                                    {12}{13}{14}{15}
                                    {12}{13}{14}{25}
                                    {12}{13}{24}{35}
                                    {12}{13}{14}{235}
                                    {12}{13}{23}{145}
                                    {12}{13}{45}{234}
                                    {12}{14}{23}{135}
                                    {12}{15}{134}{234}
                                    {15}{23}{124}{134}
                                    {15}{123}{124}{134}
                                    {15}{123}{124}{234}
                                    {12}{13}{14}{15}{23}
                                    {12}{13}{14}{23}{25}
                                    {12}{13}{14}{23}{45}
                                    {12}{13}{15}{24}{34}
                                    {12}{13}{14}{15}{234}
                                    {12}{13}{14}{25}{234}
                                    {12}{13}{14}{15}{23}{24}
                                    {12}{13}{14}{15}{23}{45}
                                    {12}{13}{14}{23}{24}{35}
                                    {15}{123}{124}{134}{234}
                                    {12}{13}{14}{15}{23}{24}{34}

Column k = 1 of A327359.
The labeled version is A327356.
Cf. A006602, A014466, A048143, A261005, A326704, A326786, A327112, A327114, A327426, A327334, A327336, A327350, A327351, A327358.

a(n > 2) = A261006(n) - A305028(n).

cp's OEIS Frontend

A327100 BII-numbers of antichains of sets with cut-connectivity 1.

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A327358 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

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A327366 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and minimum vertex-degree k.

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A327359 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

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A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1.

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A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

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A327805 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and vertex-connectivity >= k.

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A327374 BII-numbers of set-systems with vertex-connectivity 2.

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A327375 Number of set-systems with n vertices and vertex-connectivity 2.

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