OFFSET
0,53
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.
LINKS
EXAMPLE
Positions of first appearances of each integer, together with the corresponding set-systems, are:
0: {}
4: {{1,2}}
52: {{1,2},{1,3},{2,3}}
2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
vertConnSys[vts_, eds_]:=Min@@Length/@Select[Subsets[vts], Function[del, Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]
Table[vertConnSys[Union@@bpe/@bpe[n], bpe/@bpe[n]], {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 02 2019
STATUS
approved