A309326 - OEIS

%I #7 Jul 27 2019 14:57:51

%S 0,1,2,3,4,8,9,10,11,12,16,18,20,32,33,36,48,64,128,129,130,131,132,

%T 136,137,138,139,140,144,146,148,160,161,164,176,192,256,258,260,264,

%U 266,268,272,274,276,288,320,512,513,516,520,521,524,528,544,545,548

%N BII-numbers of minimal covers.

%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 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 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 The sequence of all minimal covers together with their BII-numbers begins:

%e     0: {}

%e     1: {{1}}

%e     2: {{2}}

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

%e     4: {{1,2}}

%e     8: {{3}}

%e     9: {{1},{3}}

%e    10: {{2},{3}}

%e    11: {{1},{2},{3}}

%e    12: {{1,2},{3}}

%e    16: {{1,3}}

%e    18: {{2},{1,3}}

%e    20: {{1,2},{1,3}}

%e    32: {{2,3}}

%e    33: {{1},{2,3}}

%e    36: {{1,2},{2,3}}

%e    48: {{1,3},{2,3}}

%e    64: {{1,2,3}}

%e   128: {{4}}

%e   129: {{1},{4}}

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

%t Select[Range[0,1000],And@@Table[Union@@Delete[bpe/@bpe[#],i]!=Union@@bpe/@bpe[#],{i,Length[bpe/@bpe[#]]}]&]

%Y Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

%Y Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 23 2019