

A327082


BIInumbers of setsystems with cutconnectivity 2.


12



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
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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.
We define the cutconnectivity (A326786, A327237), of a setsystem to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty setsystem, with the exception that a setsystem with one vertex and no edges has cutconnectivity 1. Except for cointersecting setsystems (A326853, A327039), this is the same as vertexconnectivity (A327334, A327051).


LINKS

Table of n, a(n) for n=1..55.


EXAMPLE

The sequence of all setsystems with cutconnectivity 2 together with their BIInumbers 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}}.


MATHEMATICA

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&]


CROSSREFS

Positions of 2's in A326786.
BIInumbers for nonspanning edgeconnectivity 2 are A327097.
BIInumbers for spanning edgeconnectivity 2 are A327108.
The cutconnectivity 1 version is A327098.
The cutconnectivity > 1 version is A327101.
Covering 2cutconnected setsystems are counted by A327112.
Covering setsystems with cutconnectivity 2 are counted by A327113.
Cf. A000120, A002218, A013922, A048793, A259862, A322387, A322388, A326031, A327041, A327114.
Sequence in context: A046300 A053738 A327101 * A154787 A233035 A250036
Adjacent sequences: A327079 A327080 A327081 * A327083 A327084 A327085


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 20 2019


STATUS

approved



