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%I #5 Nov 29 2019 01:39:30
%S 0,1,2,4,8,16,20,32,36,48,52,64,128,256,260,272,276,320,512,516,544,
%T 548,576,768,772,832,1024,1040,1056,1072,1088,2048,2064,2080,2096,
%U 2112,2304,2320,2368,2560,2592,2624,2816,2880,3072,3088,3104,3120,3136,4096
%N BII-numbers of intersecting antichains of sets.
%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 intersecting if no two edges are disjoint. It 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 2: {{2}}
%e 4: {{1,2}}
%e 8: {{3}}
%e 16: {{1,3}}
%e 20: {{1,2},{1,3}}
%e 32: {{2,3}}
%e 36: {{1,2},{2,3}}
%e 48: {{1,3},{2,3}}
%e 52: {{1,2},{1,3},{2,3}}
%e 64: {{1,2,3}}
%e 128: {{4}}
%e 256: {{1,4}}
%e 260: {{1,2},{1,4}}
%e 272: {{1,3},{1,4}}
%e 276: {{1,2},{1,3},{1,4}}
%e 320: {{1,2,3},{1,4}}
%e 512: {{2,4}}
%e 516: {{1,2},{2,4}}
%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 Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&]
%Y Intersection of A326704 (antichains) and A326910 (intersecting).
%Y Covering intersecting antichains of sets are counted by A305844.
%Y BII-numbers of antichains with empty intersection are A329560.
%Y Cf. A000120, A048143, A048793, A070939, A087086, A305857, A306007, A326031, A326361, A326912, A329628.
%K nonn
%O 1,3
%A _Gus Wiseman_, Nov 28 2019
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