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.
The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any isolated vertices) to result in a disconnected or empty set-system.
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}}
24: {{3},{1,3}}
34: {{2},{2,3}}
36: {{1,2},{2,3}}
40: {{3},{2,3}}
48: {{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}}
65: {{1},{1,2,3}}
66: {{2},{1,2,3}}
68: {{1,2},{1,2,3}}
71: {{1},{2},{1,2},{1,2,3}}
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
Positions of 2's in A326787.
BII-numbers for vertex-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for non-spanning edge-connectivity > 1 are A327102.
BII-numbers for spanning edge-connectivity 2 are A327108.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 20 2019
STATUS
approved