

A327102


BIInumbers of setsystems with nonspanning edgeconnectivity >= 2.


11



5, 6, 17, 20, 21, 24, 34, 36, 38, 40, 48, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 72, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121
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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.
A setsystem has nonspanning 2edgeconnectivity >= 2 if it is connected and any single edge can be removed (along with any noncovered vertices) without making the setsystem disconnected or empty. Alternatively, these are connected setsystems whose bridges (edges whose removal disconnects the setsystem or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).


LINKS

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


EXAMPLE

The sequence of all setsystems with nonspanning edgeconnectivity >= 2 together with their BIInumbers begins:
5: {{1},{1,2}}
6: {{2},{1,2}}
17: {{1},{1,3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
24: {{3},{1,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
40: {{3},{2,3}}
48: {{1,3},{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}}
56: {{3},{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]]]]]]]]];
edgeConn[y_]:=If[Length[csm[bpe/@y]]!=1, 0, Length[y]Max@@Length/@Select[Union[Subsets[y]], Length[csm[bpe/@#]]!=1&]];
Select[Range[0, 100], edgeConn[bpe[#]]>=2&]


CROSSREFS

Graphs with spanning edgeconnectivity >= 2 are counted by A095983.
Graphs with nonspanning edgeconnectivity >= 2 are counted by A322395.
Also positions of terms >=2 in A326787.
BIInumbers for nonspanning edgeconnectivity 2 are A327097.
BIInumbers for nonspanning edgeconnectivity 1 are A327099.
BIInumbers for spanning edgeconnectivity >= 2 are A327109.
Cf. A000120, A048793, A059166, A070939, A263296, A326031, A326749, A327076, A327101, A327102, A327108, A327148.
Sequence in context: A185508 A257338 A059013 * A191144 A327097 A035595
Adjacent sequences: A327099 A327100 A327101 * A327103 A327104 A327105


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 23 2019


STATUS

approved



