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 A330099 #12 Dec 03 2019 07:08:59
%S A330099 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 A330099 71,75,79,83,84,85,87,95,97,101,115,116,117,119,127,139,143,159,191,
%U A330099 203,207,223,255,267,271,275,276,277,279,287,307,308,309,311,319,331
%N A330099 BII-numbers of brute-force normalized set-systems.
%C A330099 First differs from A330100 in having 545 and lacking 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
%C A330099 A set-system is a finite set of finite nonempty sets of positive integers.
%C A330099 We define the brute-force 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 least representative, where the ordering of sets is first by length and then lexicographically.
%C A330099 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 A330099 Brute-force: 2067: {{1},{2},{1,3},{3,4}}
%C A330099 Lexicographic: 165: {{1},{4},{1,2},{2,3}}
%C A330099 VDD: 525: {{1},{3},{1,2},{2,4}}
%C A330099 MM: 270: {{2},{3},{1,2},{1,4}}
%C A330099 BII: 150: {{2},{4},{1,2},{1,3}}
%C A330099 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 A330099 There are A055621(n) entries m such that A326702(m) = n, where A326702(k) is the number of covered vertices in the set-system with BII-number k.
%C A330099 There are A283877(n) entries m such that A326031(m) = n, where A326031(k) is the weight of the set-system with BII-number k.
%e A330099 The sequence of all nonempty brute-force normalized set-systems together with their BII-numbers begins:
%e A330099 1: {1} 52: {12}{13}{23}
%e A330099 3: {1}{2} 53: {1}{12}{13}{23}
%e A330099 4: {12} 55: {1}{2}{12}{13}{23}
%e A330099 5: {1}{12} 63: {1}{2}{3}{12}{13}{23}
%e A330099 7: {1}{2}{12} 64: {123}
%e A330099 11: {1}{2}{3} 65: {1}{123}
%e A330099 15: {1}{2}{3}{12} 67: {1}{2}{123}
%e A330099 19: {1}{2}{13} 68: {12}{123}
%e A330099 20: {12}{13} 69: {1}{12}{123}
%e A330099 21: {1}{12}{13} 71: {1}{2}{12}{123}
%e A330099 23: {1}{2}{12}{13} 75: {1}{2}{3}{123}
%e A330099 31: {1}{2}{3}{12}{13} 79: {1}{2}{3}{12}{123}
%e A330099 33: {1}{23} 83: {1}{2}{13}{123}
%e A330099 37: {1}{12}{23} 84: {12}{13}{123}
%e A330099 51: {1}{2}{13}{23} 85: {1}{12}{13}{123}
%t A330099 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330099 brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
%t A330099 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330099 Select[Range[0,100],Sort[bpe/@bpe[#]]==brute[bpe/@bpe[#]]&]
%Y A330099 Equals the image/fixed points of the idempotent sequence A330101.
%Y A330099 Non-isomorphic multiset partitions are A007716.
%Y A330099 Unlabeled spanning set-systems by span are A055621.
%Y A330099 Unlabeled spanning set-systems by weight are A283877.
%Y A330099 Cf. A000612, A300913, A321405, A330061, A330102, A330105.
%Y A330099 Other fixed points:
%Y A330099 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330099 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330099 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330099 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330099 - BII: A330109 (set-systems).
%K A330099 nonn,eigen
%O A330099 1,3
%A A330099 _Gus Wiseman_, Dec 02 2019