This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330109 #5 Dec 06 2019 09:35:58
%S A330109 0,1,3,4,5,7,11,12,13,15,20,21,22,23,30,31,52,53,55,63,64,65,67,68,69,
%T A330109 71,75,76,77,79,84,85,86,87,94,95,116,117,119,127,139,140,141,143,148,
%U A330109 149,150,151,158,159,180,181,183,191,192,193,195,196,197,199,203
%N A330109 BII-numbers of BII-normalized set-systems.
%C A330109 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 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 A330109 We define the BII-normalization of a set-system to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the representative with the smallest BII-number.
%C A330109 For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
%C A330109 Brute-force: 2067: {{1},{2},{1,3},{3,4}}
%C A330109 Lexicographic: 165: {{1},{4},{1,2},{2,3}}
%C A330109 VDD: 525: {{1},{3},{1,2},{2,4}}
%C A330109 MM: 270: {{2},{3},{1,2},{1,4}}
%C A330109 BII: 150: {{2},{4},{1,2},{1,3}}
%e A330109 The sequence of all nonempty BII-normalized set-systems together with their BII-numbers begins:
%e A330109 1: {1} 52: {12}{13}{23}
%e A330109 3: {1}{2} 53: {1}{12}{13}{23}
%e A330109 4: {12} 55: {1}{2}{12}{13}{23}
%e A330109 5: {1}{12} 63: {1}{2}{3}{12}{13}{23}
%e A330109 7: {1}{2}{12} 64: {123}
%e A330109 11: {1}{2}{3} 65: {1}{123}
%e A330109 12: {3}{12} 67: {1}{2}{123}
%e A330109 13: {1}{3}{12} 68: {12}{123}
%e A330109 15: {1}{2}{3}{12} 69: {1}{12}{123}
%e A330109 20: {12}{13} 71: {1}{2}{12}{123}
%e A330109 21: {1}{12}{13} 75: {1}{2}{3}{123}
%e A330109 22: {2}{12}{13} 76: {3}{12}{123}
%e A330109 23: {1}{2}{12}{13} 77: {1}{3}{12}{123}
%e A330109 30: {2}{3}{12}{13} 79: {1}{2}{3}{12}{123}
%e A330109 31: {1}{2}{3}{12}{13} 84: {12}{13}{123}
%t A330109 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330109 fbi[q_]:=If[q=={},0,Total[2^q]/2];
%t A330109 biinorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],biinorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],fbi[fbi/@#]&]]];
%t A330109 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330109 Select[Range[0,100],Sort[bpe/@bpe[#]]==biinorm[bpe/@bpe[#]]&]
%Y A330109 Equals the image/fixed points of the idempotent sequence A330195.
%Y A330109 A subset of A326754.
%Y A330109 Unlabeled covering set-systems counted by vertices are A055621.
%Y A330109 Unlabeled set-systems counted by weight are A283877.
%Y A330109 BII-weight is A326031.
%Y A330109 Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A330101, A330102.
%Y A330109 Other fixed points:
%Y A330109 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330109 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330109 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330109 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330109 - BII: A330109 (set-systems).
%K A330109 nonn
%O A330109 1,3
%A A330109 _Gus Wiseman_, Dec 05 2019