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 A330123 #5 Dec 05 2019 17:42:01
%S A330123 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 A330123 71,75,77,79,84,85,87,95,97,101,109,116,117,119,127,139,143,159,173,
%U A330123 180,181,183,191,195,196,197,199,203,205,207,212,213,215,223,225,229
%N A330123 BII-numbers of MM-normalized set-systems.
%C A330123 First differs from A330110 in lacking 141 and having 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
%C A330123 A set-system is a finite set of finite nonempty set of positive integers.
%C A330123 We define the MM-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 representative with the smallest MM-number.
%C A330123 For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
%C A330123 Brute-force: 43287: {{1},{2,3},{2,2,4}}
%C A330123 Lexicographic: 43143: {{1},{2,4},{2,2,3}}
%C A330123 VDD: 15515: {{2},{1,3},{1,1,4}}
%C A330123 MM: 15265: {{2},{1,4},{1,1,3}}
%C A330123 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 A330123 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%e A330123 The sequence of all MM-normalized set-systems together with their BII-numbers begins:
%e A330123 0: {} 45: {{1},{3},{1,2},{2,3}}
%e A330123 1: {{1}} 52: {{1,2},{1,3},{2,3}}
%e A330123 3: {{1},{2}} 53: {{1},{1,2},{1,3},{2,3}}
%e A330123 4: {{1,2}} 55: {{1},{2},{1,2},{1,3},{2,3}}
%e A330123 5: {{1},{1,2}} 63: {{1},{2},{3},{1,2},{1,3},{2,3}}
%e A330123 7: {{1},{2},{1,2}} 64: {{1,2,3}}
%e A330123 11: {{1},{2},{3}} 65: {{1},{1,2,3}}
%e A330123 13: {{1},{3},{1,2}} 67: {{1},{2},{1,2,3}}
%e A330123 15: {{1},{2},{3},{1,2}} 68: {{1,2},{1,2,3}}
%e A330123 20: {{1,2},{1,3}} 69: {{1},{1,2},{1,2,3}}
%e A330123 21: {{1},{1,2},{1,3}} 71: {{1},{2},{1,2},{1,2,3}}
%e A330123 23: {{1},{2},{1,2},{1,3}} 75: {{1},{2},{3},{1,2,3}}
%e A330123 31: {{1},{2},{3},{1,2},{1,3}} 77: {{1},{3},{1,2},{1,2,3}}
%e A330123 33: {{1},{2,3}} 79: {{1},{2},{3},{1,2},{1,2,3}}
%e A330123 37: {{1},{1,2},{2,3}} 84: {{1,2},{1,3},{1,2,3}}
%t A330123 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A330123 mmnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],mmnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[SortBy[brute[m,1],Map[Times@@Prime/@#&,#,{0,1}]&]]];
%t A330123 brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330123 Select[Range[0,100],Sort[bpe/@bpe[#]]==mmnorm[bpe/@bpe[#]]&]
%Y A330123 A subset of A326754.
%Y A330123 Non-isomorphic multiset partitions are A007716.
%Y A330123 Unlabeled spanning set-systems counted by vertices are A055621.
%Y A330123 Unlabeled set-systems counted by weight are A283877.
%Y A330123 MM-weight is A302242.
%Y A330123 Cf. A000120, A000612, A048793, A056239, A070939, A112798, A300913, A319559, A320456, A326031, A330101, A330102, A330194.
%Y A330123 Other fixed points:
%Y A330123 - Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
%Y A330123 - Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
%Y A330123 - VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
%Y A330123 - MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
%Y A330123 - BII: A330109 (set-systems).
%K A330123 nonn
%O A330123 1,3
%A A330123 _Gus Wiseman_, Dec 05 2019