A327051 -id:A327051 - cp's OEIS frontend

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

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.

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}]

A327234 Smallest BII-number of a set-system with cut-connectivity n.

Original entry on oeis.org

0, 1, 4, 52, 2868

Offset: 0

Gus Wiseman, Sep 03 2019

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) 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), this is the same as vertex-connectivity (A327051).
Conjecture: a(n > 1) = A327373(n) = the BII-number of K_n.

			The sequence of terms together with their corresponding set-systems:
     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}}

The same for spanning edge-connectivity is A327147.
The cut-connectivity of the set-system with BII-number n is A326786(n).
Cf. A000120, A002450, A029931, A048793, A070939, A259862, A326031, A327082, A327098, A327125, A327126, A327127, A327373.

A327376 BII-numbers of set-systems with vertex-connectivity 3.

Original entry on oeis.org

2868, 2869, 2870, 2871, 2876, 2877, 2878, 2879, 2880, 2881, 2882, 2883, 2884, 2885, 2886, 2887, 2888, 2889, 2890, 2891, 2892, 2893, 2894, 2895, 2896, 2897, 2898, 2899, 2900, 2901, 2902, 2903, 2904, 2905, 2906, 2907, 2908, 2909, 2910, 2911, 2912, 2913, 2914

Offset: 1

Gus Wiseman, Sep 05 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 3 together with their BII-numbers begins:
  2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2869: {{1},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2870: {{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2871: {{1},{2},{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2876: {{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2877: {{1},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2878: {{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2879: {{1},{2},{1,2},{3},{1,3},{2,3},{1,4},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}
  2881: {{1},{1,2,3},{1,4},{2,4},{3,4}}
  2882: {{2},{1,2,3},{1,4},{2,4},{3,4}}
  2883: {{1},{2},{1,2,3},{1,4},{2,4},{3,4}}
  2884: {{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2885: {{1},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2886: {{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2887: {{1},{2},{1,2},{1,2,3},{1,4},{2,4},{3,4}}
  2888: {{3},{1,2,3},{1,4},{2,4},{3,4}}
  2889: {{1},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2890: {{2},{3},{1,2,3},{1,4},{2,4},{3,4}}
  2891: {{1},{2},{3},{1,2,3},{1,4},{2,4},{3,4}}

Positions of 3's in A327051.
BII-numbers for vertex-connectivity 2 are A327374.
BII-numbers for spanning edge-connectivity >= 3 are A327110.
The enumeration of labeled graphs by vertex-connectivity is A327334.
Cf. A000120, A013922, A048793, A070939, A259862, A323818, A326031, A326753, A326786, A327375.

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[0,3000],vertConnSys[Union@@bpe/@bpe[#],bpe/@bpe[#]]==3&]

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A327128 Number of set-systems with n vertices whose edge-set has cut-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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A327234 Smallest BII-number of a set-system with cut-connectivity n.

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A327376 BII-numbers of set-systems with vertex-connectivity 3.

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