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 A368094 #21 Dec 28 2023 11:32:09
%S A368094 0,0,0,0,1,1,5,12,36,97,291
%N A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.
%C A368094 A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.
%C A368094 The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
%H A368094 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368094 Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:
%e A368094 {{1},{2},{3},{2,3}} {{1},{2},{1,3},{2,3}} {{1},{2},{1,2},{3,4,5}}
%e A368094 {{1},{2},{3},{1,2,3}} {{1},{3},{2,3},{1,2,3}}
%e A368094 {{2},{3},{1,3},{2,3}} {{1},{4},{1,4},{2,3,4}}
%e A368094 {{3},{4},{1,2},{3,4}} {{2},{3},{2,3},{1,2,3}}
%e A368094 {{1},{2},{3},{4},{3,4}} {{3},{1,2},{1,3},{2,3}}
%e A368094 {{1},{2},{3},{1,3},{2,3}}
%e A368094 {{1},{2},{3},{2,4},{3,4}}
%e A368094 {{1},{2},{3},{4},{2,3,4}}
%e A368094 {{1},{3},{4},{2,4},{3,4}}
%e A368094 {{1},{4},{5},{2,3},{4,5}}
%e A368094 {{2},{3},{4},{1,2},{3,4}}
%e A368094 {{1},{2},{3},{4},{5},{4,5}}
%t A368094 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A368094 mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368094 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368094 Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]
%Y A368094 The case of unlabeled graphs is A140637, complement A134964.
%Y A368094 The case of labeled graphs is A367867, complement A133686.
%Y A368094 The labeled version is A367903, ranks A367907.
%Y A368094 The complement is counted by A368095, connected A368410.
%Y A368094 Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
%Y A368094 Minimal multiset partitions of this type are ranked by A368187.
%Y A368094 The connected case is A368409.
%Y A368094 Factorizations of this type are counted by A368413, complement A368414.
%Y A368094 Allowing repeated edges gives A368421, complement A368422.
%Y A368094 A000110 counts set partitions, non-isomorphic A000041.
%Y A368094 A003465 counts covering set-systems, unlabeled A055621.
%Y A368094 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368094 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368094 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368094 Cf. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.
%K A368094 nonn,more
%O A368094 0,7
%A A368094 _Gus Wiseman_, Dec 23 2023