cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A327363 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 8, 4, 1, 0, 64, 38, 10, 1, 0, 1024, 728, 238, 26, 1, 0
Offset: 0

Views

Author

Gus Wiseman, Sep 26 2019

Keywords

Comments

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.

Examples

			Triangle begins:
     1
     1    0
     2    1    0
     8    4    1    0
    64   38   10    1    0
  1024  728  238   26    1    0

Crossrefs

Column k = 0 is A006125.
Column k = 1 is A001187.
Column k = 2 is A013922.
The unlabeled version is A327805.
Row-wise partial sums of A327334 (vertex-connectivity exactly k).
Cf. A259862, A327114, A327125, A327126, A327127, A327806.

Programs

  • Mathematica
    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],#]>=k&]],{n,0,4},{k,0,n}]

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

Views

Author

Gus Wiseman, Sep 05 2019

Keywords

Comments

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.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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&]

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 9, 3, 2, 0, 29, 14, 10, 6, 0, 209, 157, 128, 91, 54, 0
Offset: 0

Views

Author

Gus Wiseman, Sep 26 2019

Keywords

Comments

An antichain is a set of sets, none of which is a subset of any other.
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.

Examples

			Triangle begins:
    1
    2   0
    4   1   0
    9   3   2   0
   29  14  10   6   0
  209 157 128  91  54   0

Crossrefs

Column k = 0 is A306505.
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.
Except for the first column, same as A327358 (the covering case).
The labeled version is A327806.
Cf. A014466, A293606, A326704, A326950, A327062, A327334, A327350, A327351, A327805, A327808.
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