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 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.
A set-system has non-spanning 2-edge-connectivity >= 2 if it is connected and any single edge can be removed (along with any non-covered vertices) without making the set-system disconnected or empty. Alternatively, these are connected set-systems whose bridges (edges whose removal disconnects the set-system or leaves isolated vertices) are all endpoints (edges intersecting only one other edge).
EXAMPLE
The sequence of all set-systems with non-spanning edge-connectivity >= 2 together with their BII-numbers 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 edge-connectivity >= 2 are counted by A095983.
Graphs with non-spanning edge-connectivity >= 2 are counted by A322395.
Also positions of terms >=2 in A326787.
BII-numbers for non-spanning edge-connectivity 2 are A327097.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2019
STATUS
approved