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 A330100 #6 Dec 05 2019 08:23:38
%S A330100 0,1,3,4,5,7,11,15,19,20,21,23,31,33,37,51,52,53,55,63,64,65,67,68,69,
%T A330100 71,75,79,83,84,85,87,95,97,101,115,116,117,119,127,139,143,159,179,
%U A330100 180,181,183,191,203,207,211,212,213,215,223,225,229,243,244,245,247
%N A330100 BII-numbers of VDD-normalized set-systems.
%C A330100 First differs from A330099 in lacking 545 and having 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
%C A330100 A set-system is a finite set of finite nonempty sets of positive integers.
%C A330100 We define the VDD (vertex-degrees decreasing) 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, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
%C A330100 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 A330100 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 A330100 Brute-force: 2067: {{1},{2},{1,3},{3,4}}
%C A330100 Lexicographic: 165: {{1},{4},{1,2},{2,3}}
%C A330100 VDD: 525: {{1},{3},{1,2},{2,4}}
%C A330100 MM: 270: {{2},{3},{1,2},{1,4}}
%C A330100 BII: 150: {{2},{4},{1,2},{1,3}}
%e A330100 The sequence of all nonempty VDD-normalized set-systems together with their BII-numbers begins:
%e A330100 1: {1} 52: {12}{13}{23}
%e A330100 3: {1}{2} 53: {1}{12}{13}{23}
%e A330100 4: {12} 55: {1}{2}{12}{13}{23}
%e A330100 5: {1}{12} 63: {1}{2}{3}{12}{13}{23}
%e A330100 7: {1}{2}{12} 64: {123}
%e A330100 11: {1}{2}{3} 65: {1}{123}
%e A330100 15: {1}{2}{3}{12} 67: {1}{2}{123}
%e A330100 19: {1}{2}{13} 68: {12}{123}
%e A330100 20: {12}{13} 69: {1}{12}{123}
%e A330100 21: {1}{12}{13} 71: {1}{2}{12}{123}
%e A330100 23: {1}{2}{12}{13} 75: {1}{2}{3}{123}
%e A330100 31: {1}{2}{3}{12}{13} 79: {1}{2}{3}{12}{123}
%e A330100 33: {1}{23} 83: {1}{2}{13}{123}
%e A330100 37: {1}{12}{23} 84: {12}{13}{123}
%e A330100 51: {1}{2}{13}{23} 85: {1}{12}{13}{123}
%t A330100 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330100 sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
%t A330100 sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
%t A330100 Select[Range[0,100],Sort[bpe/@bpe[#]]==sysnorm[bpe/@bpe[#]]&]
%Y A330100 Equals the image/fixed points of the idempotent sequence A330102.
%Y A330100 A subset of A326754.
%Y A330100 Non-isomorphic multiset partitions are A007716.
%Y A330100 Unlabeled spanning set-systems counted by vertices are A055621.
%Y A330100 Unlabeled set-systems counted by weight are A283877.
%Y A330100 Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A330061, A330101.
%Y A330100 Other fixed points:
%Y A330100 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330100 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330100 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330100 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330100 - BII: A330109 (set-systems).
%K A330100 nonn,eigen
%O A330100 0,3
%A A330100 _Gus Wiseman_, Dec 04 2019