A329625 - OEIS

%I #5 Nov 29 2019 01:39:38

%S 0,1,5,7,23,31,63,127,383,511,1023,2047,4095,8191

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

%C 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 The sequence of terms together with their corresponding set-systems begins:

%e      0: {}

%e      1: {{1}}

%e      5: {{1},{1,2}}

%e      7: {{1},{2},{1,2}}

%e     23: {{1},{2},{1,2},{1,3}}

%e     31: {{1},{2},{1,2},{3},{1,3}}

%e     63: {{1},{2},{1,2},{3},{1,3},{2,3}}

%e    127: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3}}

%e    383: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{1,4}}

%e    511: {{1},{2},{1,2},{3},{1,3},{2,3},{1,2,3},{4},{1,4}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t 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 First/@GatherBy[Select[Range[0,1000],Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]

%Y The smallest BII-number of a set-system with n edges is A000225(n).

%Y The smallest BII-number of a set-system with n vertices is A072639(n).

%Y BII-numbers of connected set-systems are A326749.

%Y MM-numbers of connected set-systems are A328514.

%Y The case of clutters is A329627.

%Y Cf. A048143, A048793, A070939, A304716, A305078, A326031, A326753, A329552, A329553, A329626.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Nov 28 2019