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 A368409 #7 Dec 26 2023 08:32:03
%S A368409 0,0,0,0,1,0,3,5,16,41,130
%N A368409 Number of non-isomorphic connected set-systems of weight n contradicting a strict version of the axiom of choice.
%C A368409 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 A368409 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 A368409 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368409 Non-isomorphic representatives of the a(4) = 1 through a(8) = 16 set-systems:
%e A368409 {1}{2}{12} . {1}{2}{13}{23} {1}{3}{23}{123} {1}{5}{15}{2345}
%e A368409 {1}{2}{3}{123} {1}{4}{14}{234} {2}{13}{23}{123}
%e A368409 {2}{3}{13}{23} {2}{3}{23}{123} {3}{13}{23}{123}
%e A368409 {3}{12}{13}{23} {3}{4}{34}{1234}
%e A368409 {1}{2}{3}{13}{23} {1}{2}{13}{24}{34}
%e A368409 {1}{2}{3}{14}{234}
%e A368409 {1}{2}{3}{23}{123}
%e A368409 {1}{2}{3}{4}{1234}
%e A368409 {1}{3}{4}{14}{234}
%e A368409 {2}{3}{12}{13}{23}
%e A368409 {2}{3}{13}{24}{34}
%e A368409 {2}{3}{14}{24}{34}
%e A368409 {2}{3}{4}{14}{234}
%e A368409 {2}{4}{13}{24}{34}
%e A368409 {3}{4}{13}{24}{34}
%e A368409 {3}{4}{14}{24}{34}
%t A368409 sps[{}]:={{}}; sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368409 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2, {#1}]&,#]]&/@IntegerPartitions[n]}];
%t A368409 brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];
%t A368409 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
%t A368409 Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]
%Y A368409 For unlabeled graphs we have A140636, connected case of A140637.
%Y A368409 For labeled graphs: A140638, connected case of A367867 (complement A133686).
%Y A368409 This is the connected case of A368094.
%Y A368409 The complement is A368410, connected case of A368095.
%Y A368409 Allowing repeats: A368411, connected case of A368097, ranks A355529.
%Y A368409 Complement with repeats: A368412, connected case of A368098, ranks A368100.
%Y A368409 Allowing repeat edges only: connected case of A368421 (complement A368422).
%Y A368409 A000110 counts set partitions, non-isomorphic A000041.
%Y A368409 A003465 counts covering set-systems, unlabeled A055621.
%Y A368409 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368409 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368409 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368409 Cf. A134964, A302545, A306005, A317533, A321405, A326031, A367903, A367907, A368187, A368413.
%K A368409 nonn,more
%O A368409 0,7
%A A368409 _Gus Wiseman_, Dec 25 2023