A327097 -id:A327097 - cp's OEIS frontend

A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Aug 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 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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{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}}
  32: {{2,3}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
BII-numbers for vertex-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
Graphs with spanning edge-connectivity 1 are counted by A327145.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==1&]

A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 848, 849, 850

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...
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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-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}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of 2's in A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.

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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,100],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]==2&]

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any isolated vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   1
   1   1
   1   3   3   1
   4  18  27  14   1
  56 250 402 240  65  10   1

Row sums are A006125.
Column k = 0 is A327199.
Column k = 1 is A327231.
The corresponding triangle for vertex-connectivity is A327125.
The corresponding triangle for spanning edge-connectivity is A327069.
The covering version is A327149.
The unlabeled version is A327236, with covering version A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.

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]]]]]]]]];
edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.

a(20)-a(28) from Robert Price, May 25 2021

A327099 BII-numbers of set-systems with non-spanning edge-connectivity 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 16, 22, 23, 25, 28, 29, 30, 31, 32, 37, 39, 42, 44, 45, 46, 47, 49, 50, 51, 57, 58, 59, 64, 67, 73, 74, 75, 76, 77, 78, 79, 82, 83, 90, 91, 97, 99, 105, 107, 128, 256, 262, 263, 278, 279, 280, 281, 284, 285, 286, 287, 292, 293, 294, 295, 300

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.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to result in a disconnected or empty set-system.

			The sequence of all set-systems with non-spanning edge-connectivity 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{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}}
  32: {{2,3}}
  37: {{1},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
  42: {{2},{3},{2,3}}
  44: {{1,2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  46: {{2},{1,2},{3},{2,3}}

Positions of 1's in A326787.
Simple graphs with non-spanning edge-connectivity 1 are A327071.
BII-numbers for non-spanning edge-connectivity >= 1 are A326749.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for spanning edge-connectivity 1 are A327111.
BII-numbers for vertex-connectivity 1 are A327114.
Covering set-systems with non-spanning edge-connectivity 1 are counted by A327129.
Cf. A048793, A052446, A070939, A322338, A326031, 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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]==1&]

A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2.

Original entry on oeis.org

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826

Offset: 1

Gus Wiseman, Aug 23 2019

nonn

Differs from A327108 in having 116, 117, 118, 119, 124, 125, 126, 127, ...
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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty set-system.

			The sequence of all set-systems with spanning edge-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}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

Positions of terms >= 2 in  A327144.
Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with spanning edge-connectivity 2 are counted by A327146.
Set-systems with spanning edge-connectivity 2 are counted by A327130.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity 2 are A327108.
BII-numbers for spanning edge-connectivity 1 are A327111.
Cf. A326749, A326753, A326787, A327041, A327069, A327071, A327075.

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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Select[Range[0,1000],spanEdgeConn[Union@@bpe/@bpe[#],bpe/@bpe[#]]>=2&]

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&]

A327102 BII-numbers of set-systems with non-spanning edge-connectivity >= 2.

Original entry on oeis.org

5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Aug 23 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 has non-spanning 2-edge-connectivity >= 2 if it is connected and any single edge can be removed (along with any non-covered vertices) without making the set-system disconnected or empty. Alternatively, these are connected set-systems whose bridges (edges whose removal disconnects the set-system or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).

			The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers begins:
   5: {{1},{1,2}}
   6: {{2},{1,2}}
  17: {{1},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  24: {{3},{1,3}}
  34: {{2},{2,3}}
  36: {{1,2},{2,3}}
  38: {{2},{1,2},{2,3}}
  40: {{3},{2,3}}
  48: {{1,3},{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}}
  56: {{3},{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}}

Graphs with spanning edge-connectivity >= 2 are counted by A095983.
Graphs with non-spanning edge-connectivity >= 2 are counted by A322395.
Also positions of terms >=2 in A326787.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.

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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1,0,Length[y]-Max@@Length/@Select[Union[Subsets[y]],Length[csm[bpe/@#]]!=1&]];
Select[Range[0,100],edgeConn[bpe[#]]>=2&]

A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 1, 4, 5, 10, 8, 5, 1, 1

Offset: 0

Gus Wiseman, Sep 03 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a disconnected or empty graph, ignoring isolated vertices.

			Triangle begins:
  1
  1
  1  1
  1  1  1  1
  2  2  3  3  1
  4  5 10  8  5  1  1

Gus Wiseman, Unlabeled graphs with 5 vertices, organized by non-spanning edge-connectivity (isolated vertices not shown).

Row sums are A000088.
Column k = 0 is A327235.
The labeled version is A327148.
The covering version is A327201.
Spanning edge-connectivity is A263296.
Vertex-connectivity is A259862.
Cf. A322338, A322396, A326787, A327069, A327077, A327097, A327099, A327102, A327200, A327231.

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]]]]]]]]];
edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]]],{n,0,5},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

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).

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 3, 12, 15, 10, 1, 40, 180, 297, 180, 60, 10, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   {}
   0   1
   0   0   3   1
   3  12  15  10   1
  40 180 297 180  60  10   1

Row sums are A006129.
Column k = 0 is A327070.
Column k = 1 is A327079.
The corresponding triangle for vertex-connectivity is A327126.
The corresponding triangle for spanning edge-connectivity is A327069.
The non-covering version is A327148.
The unlabeled version is A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.

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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.

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A327111 BII-numbers of set-systems with spanning edge-connectivity 1.

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A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

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A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

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A327099 BII-numbers of set-systems with non-spanning edge-connectivity 1.

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A327109 BII-numbers of set-systems with spanning edge-connectivity >= 2.

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

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A327102 BII-numbers of set-systems with non-spanning edge-connectivity >= 2.

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A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

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

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