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%I #7 Jul 27 2019 14:57:51
%S 0,4,16,20,32,36,48,52,256,260,272,276,288,292,304,308,512,516,528,
%T 532,544,548,560,564,768,772,784,788,800,804,816,820,2048,2052,2064,
%U 2068,2080,2084,2096,2100,2304,2308,2320,2324,2336,2340,2352,2356,2560,2564
%N BII-numbers of simple labeled graphs.
%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. Elements of a set-system are sometimes called edges.
%C Also numbers whose binary indices all belong to A018900.
%e The sequence of all simple labeled graphs together with their BII-numbers begins:
%e 0: {}
%e 4: {{1,2}}
%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 256: {{1,4}}
%e 260: {{1,2},{1,4}}
%e 272: {{1,3},{1,4}}
%e 276: {{1,2},{1,3},{1,4}}
%e 288: {{2,3},{1,4}}
%e 292: {{1,2},{2,3},{1,4}}
%e 304: {{1,3},{2,3},{1,4}}
%e 308: {{1,2},{1,3},{2,3},{1,4}}
%e 512: {{2,4}}
%e 516: {{1,2},{2,4}}
%e 528: {{1,3},{2,4}}
%e 532: {{1,2},{1,3},{2,4}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,100],SameQ[2,##]&@@Length/@bpe/@bpe[#]&]
%Y Cf. A000120, A006125, A006129, A018900, A048793, A062880, A070939, A309356 (same for MM-numbers), A322551, 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 1,2
%A _Gus Wiseman_, Jul 25 2019
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