

A327374


BIInumbers of setsystems with vertexconnectivity 2.


2



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
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OFFSET

1,1


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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
The vertexconnectivity of a setsystem is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a nonconnected setsystem or singleton. Note that this means a single node has vertexconnectivity 0.


LINKS

Table of n, a(n) for n=1..61.


EXAMPLE

The sequence of all setsystems with vertexconnectivity 2 together with their BIInumbers 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}}


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]1csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]];
Select[Range[0, 200], vertConnSys[Union@@bpe/@bpe[#], bpe/@bpe[#]]==2&]


CROSSREFS

Positions of 2's in A327051.
Cutconnectivity 2 is A327082.
Spanning edgeconnectivity 2 is A327108.
Nonspanning edgeconnectivity 2 is A327097.
Vertexconnectivity 3 is A327376.
Labeled graphs with vertexconnectivity 2 are A327198.
Setsystems with vertexconnectivity 2 are A327375.
The enumeration of labeled graphs by vertexconnectivity is A327334.
Cf. A000120, A013922, A048793, A070939, A259862, A326031, A326749, A327336.
Sequence in context: A143723 A252713 A249404 * A327109 A327108 A295156
Adjacent sequences: A327371 A327372 A327373 * A327375 A327376 A327377


KEYWORD

nonn


AUTHOR

Gus Wiseman, Sep 04 2019


STATUS

approved



