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A309326 BII-numbers of minimal covers.

%I A309326 #7 Jul 27 2019 14:57:51
%S A309326 0,1,2,3,4,8,9,10,11,12,16,18,20,32,33,36,48,64,128,129,130,131,132,
%T A309326 136,137,138,139,140,144,146,148,160,161,164,176,192,256,258,260,264,
%U A309326 266,268,272,274,276,288,320,512,513,516,520,521,524,528,544,545,548
%N A309326 BII-numbers of minimal covers.
%C A309326 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 finite set of finite nonempty sets 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.
%C A309326 Elements of a set-system are sometimes called edges. A minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.
%e A309326 The sequence of all minimal covers together with their BII-numbers begins:
%e A309326     0: {}
%e A309326     1: {{1}}
%e A309326     2: {{2}}
%e A309326     3: {{1},{2}}
%e A309326     4: {{1,2}}
%e A309326     8: {{3}}
%e A309326     9: {{1},{3}}
%e A309326    10: {{2},{3}}
%e A309326    11: {{1},{2},{3}}
%e A309326    12: {{1,2},{3}}
%e A309326    16: {{1,3}}
%e A309326    18: {{2},{1,3}}
%e A309326    20: {{1,2},{1,3}}
%e A309326    32: {{2,3}}
%e A309326    33: {{1},{2,3}}
%e A309326    36: {{1,2},{2,3}}
%e A309326    48: {{1,3},{2,3}}
%e A309326    64: {{1,2,3}}
%e A309326   128: {{4}}
%e A309326   129: {{1},{4}}
%t A309326 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A309326 Select[Range[0,1000],And@@Table[Union@@Delete[bpe/@bpe[#],i]!=Union@@bpe/@bpe[#],{i,Length[bpe/@bpe[#]]}]&]
%Y A309326 Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.
%Y A309326 Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
%K A309326 nonn
%O A309326 1,3
%A A309326 _Gus Wiseman_, Jul 23 2019