

A327101


BIInumbers of 2cutconnected setsystems (cutconnectivity >= 2).


10



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
A setsystem is 2cutconnected if any single vertex can be removed (along with any empty edges) without making the setsystem disconnected or empty. Except for cointersecting setsystems (A326853), this is the same as 2vertexconnectivity.


LINKS

Table of n, a(n) for n=1..64.


FORMULA

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).


EXAMPLE

The sequence of all 2cutconnected setsystems 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}}
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}}


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]]]]]]]]];
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&]


CROSSREFS

Positions of numbers >= 2 in A326786.
2cutconnected graphs are counted by A013922, if we assume A013922(2) = 0.
2cutconnected integer partitions are counted by A322387.
BIInumbers for cutconnectivity 2 are A327082.
BIInumbers for cutconnectivity 1 are A327098.
BIInumbers for nonspanning edgeconnectivity >= 2 are A327102.
BIInumbers for spanning edgeconnectivity >= 2 are A327109.
Covering 2cutconnected setsystems are counted by A327112.
Covering setsystems with cutconnectivity 2 are counted by A327113.
The labeled cutconnectivity triangle is A327125, with unlabeled version A327127.
Cf. A000120, A002218, A048793, A070939, A259862, A322388, A326031, A326749, A327097, A327108.
Sequence in context: A294228 A046300 A053738 * A327082 A154787 A233035
Adjacent sequences: A327098 A327099 A327100 * A327102 A327103 A327104


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 22 2019


STATUS

approved



