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%I #11 Jan 06 2020 18:09:45
%S 0,1,20,52,308,820,2868,68404,199476,723764
%N Smallest BII-number of a clutter (connected antichain) 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.
%C A set-system is an antichain if no edge is a proper subset of any other.
%C For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.
%e The sequence of terms together with their corresponding set-systems begins:
%e 0: {}
%e 1: {{1}}
%e 20: {{1,2},{1,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 308: {{1,2},{1,3},{2,3},{1,4}}
%e 820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
%e 2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%e 68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
%e 199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
%e 723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%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[0,10000],stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]
%Y The version for MM-numbers is A329555.
%Y BII-numbers of clutters are A326750.
%Y Clutters of sets are counted by A048143.
%Y Minimum BII-numbers of connected set-systems are A329625.
%Y Minimum BII-numbers of antichains are A329626.
%Y MM-numbers of connected weak antichains of multisets are A329559.
%Y Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 28 2019
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