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 is 2-cut-connected if any single vertex can be removed (along with any empty edges) without making the set-system disconnected or empty. Except for cointersecting set-systems (A326853), this is the same as 2-vertex-connectivity.
FORMULA
EXAMPLE
The sequence of all 2-cut-connected set-systems together with their BII-numbers 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.
2-cut-connected integer partitions are counted by A322387.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for cut-connectivity 1 are A327098.
BII-numbers for non-spanning edge-connectivity >= 2 are A327102.
BII-numbers for spanning edge-connectivity >= 2 are A327109.
Covering 2-cut-connected set-systems are counted by A327112.
Covering set-systems with cut-connectivity 2 are counted by A327113.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 22 2019
STATUS
approved