OFFSET
1,2
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.
We define the cut-connectivity (A326786, A327237), of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex and no edges has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039), this is the same as vertex-connectivity (A327334, A327051).
EXAMPLE
The sequence of all set-systems with cut-connectivity 1 together with their BII-numbers begins:
1: {{1}}
2: {{2}}
8: {{3}}
20: {{1,2},{1,3}}
21: {{1},{1,2},{1,3}}
22: {{2},{1,2},{1,3}}
23: {{1},{2},{1,2},{1,3}}
28: {{1,2},{3},{1,3}}
29: {{1},{1,2},{3},{1,3}}
30: {{2},{1,2},{3},{1,3}}
31: {{1},{2},{1,2},{3},{1,3}}
36: {{1,2},{2,3}}
37: {{1},{1,2},{2,3}}
38: {{2},{1,2},{2,3}}
39: {{1},{2},{1,2},{2,3}}
44: {{1,2},{3},{2,3}}
45: {{1},{1,2},{3},{2,3}}
46: {{2},{1,2},{3},{2,3}}
47: {{1},{2},{1,2},{3},{2,3}}
48: {{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]]]]]]]]];
vertConnSys[sys_]:=If[Length[csm[sys]]!=1, 0, Min@@Length/@Select[Subsets[Union@@sys], Function[del, Length[csm[DeleteCases[DeleteCases[sys, Alternatives@@del, {2}], {}]]]!=1]]];
Select[Range[0, 100], vertConnSys[bpe/@bpe[#]]==1&]
CROSSREFS
A subset of A326749.
Positions of 1's in A326786.
BII-numbers for cut-connectivity 2 are A327082.
BII-numbers for non-spanning edge-connectivity 1 are A327099.
BII-numbers for spanning edge-connectivity 1 are A327111.
Integer partitions with cut-connectivity 1 are counted by A322390.
Labeled connected separable graphs are counted by A327114.
Connected separable set-systems are counted by A327197.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 21 2019
STATUS
approved