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 A330110 #5 Dec 06 2019 09:36:05
%S A330110 0,1,3,4,5,7,11,13,15,20,21,23,31,33,37,45,52,53,55,63,64,65,67,68,69,
%T A330110 71,75,77,79,84,85,87,95,97,101,109,116,117,119,127,139,141,143,149,
%U A330110 151,159,165,173,181,183,191,193,195,197,199,203,205,207,213,215
%N A330110 BII-numbers of lexicographically normalized set-systems.
%C A330110 First differs from A330099 in having 13 and lacking 19.
%C A330110 First differs from A330123 in having 141 and lacking 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
%C A330110 We define the lexicographic normalization of a multiset of multisets 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 lexicographically least of these representatives.
%C A330110 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 A330110 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 A330110 Brute-force: 2067: {{1},{2},{1,3},{3,4}}
%C A330110 Lexicographic: 165: {{1},{4},{1,2},{2,3}}
%C A330110 VDD: 525: {{1},{3},{1,2},{2,4}}
%C A330110 MM: 270: {{2},{3},{1,2},{1,4}}
%C A330110 BII: 150: {{2},{4},{1,2},{1,3}}
%e A330110 The sequence of all nonempty lexicographically normalized set-systems together with their BII-numbers begins:
%e A330110 1: {1} 52: {12}{13}{23}
%e A330110 3: {1}{2} 53: {1}{12}{13}{23}
%e A330110 4: {12} 55: {1}{2}{12}{13}{23}
%e A330110 5: {1}{12} 63: {1}{2}{3}{12}{13}{23}
%e A330110 7: {1}{2}{12} 64: {123}
%e A330110 11: {1}{2}{3} 65: {1}{123}
%e A330110 13: {1}{3}{12} 67: {1}{2}{123}
%e A330110 15: {1}{2}{3}{12} 68: {12}{123}
%e A330110 20: {12}{13} 69: {1}{12}{123}
%e A330110 21: {1}{12}{13} 71: {1}{2}{12}{123}
%e A330110 23: {1}{2}{12}{13} 75: {1}{2}{3}{123}
%e A330110 31: {1}{2}{3}{12}{13} 77: {1}{3}{12}{123}
%e A330110 33: {1}{23} 79: {1}{2}{3}{12}{123}
%e A330110 37: {1}{12}{23} 84: {12}{13}{123}
%e A330110 45: {1}{3}{12}{23} 85: {1}{12}{13}{123}
%Y A330110 A subset of A326754.
%Y A330110 Unlabeled covering set-systems counted by vertices are A055621.
%Y A330110 Unlabeled set-systems counted by weight are A283877.
%Y A330110 BII-weight is A326031.
%Y A330110 Cf. A000120, A000612, A048793, A070939, A300913, A319559, A330101, A330102, A330194, A330195.
%Y A330110 Other fixed points:
%Y A330110 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330110 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330110 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330110 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330110 - BII: A330109 (set-systems).
%K A330110 nonn
%O A330110 1,3
%A A330110 _Gus Wiseman_, Dec 05 2019