cp's OEIS Frontend

A329625 Smallest BII-number of a connected set-system with n edges.

%I A329625 #5 Nov 29 2019 01:39:38
%S A329625 0,1,5,7,23,31,63,127,383,511,1023,2047,4095,8191
%N A329625 Smallest BII-number of a connected set-system with n edges.
%C A329625 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 of positive integers) 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.
%e A329625 The sequence of terms together with their corresponding set-systems begins:
%e A329625      0: {}
%e A329625      1: {{1}}
%e A329625      5: {{1},{1,2}}
%e A329625      7: {{1},{2},{1,2}}
%e A329625     23: {{1},{2},{1,2},{1,3}}
%e A329625     31: {{1},{2},{1,2},{3},{1,3}}
%e A329625     63: {{1},{2},{1,2},{3},{1,3},{2,3}}
%e A329625    127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}
%e A329625    383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}
%e A329625    511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}
%t A329625 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A329625 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]]]]]]]]];
%t A329625 First/@GatherBy[Select[Range[0,1000],Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]
%Y A329625 The smallest BII-number of a set-system with n edges is A000225(n).
%Y A329625 The smallest BII-number of a set-system with n vertices is A072639(n).
%Y A329625 BII-numbers of connected set-systems are A326749.
%Y A329625 MM-numbers of connected set-systems are A328514.
%Y A329625 The case of clutters is A329627.
%Y A329625 Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.
%K A329625 nonn,more
%O A329625 0,3
%A A329625 _Gus Wiseman_, Nov 28 2019