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%I #5 Nov 29 2019 01:39:46
%S 0,1,3,11,139,820,2868,35636,199476,723764
%N Smallest BII-number of an 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.
%e The sequence of terms together with their corresponding set-systems begins:
%e 0: {}
%e 1: {{1}}
%e 3: {{1},{2}}
%e 11: {{1},{2},{3}}
%e 139: {{1},{2},{3},{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 35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{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_]:=!Apply[Or,Outer[#1=!=#2&&SubsetQ[#1,#2]&,u,u,1],{0,1}];
%t First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&],Length[bpe[#]]&]
%Y The connected case is A329627.
%Y The intersecting case is A329628.
%Y BII-numbers of antichains are A326704.
%Y Antichain covers are A006126.
%Y Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 28 2019
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