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 A368422 #6 Dec 27 2023 09:41:11
%S A368422 1,1,2,4,9,18,43,95,233,569
%N A368422 Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.
%C A368422 A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.
%C A368422 The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.
%H A368422 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368422 Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 set multipartitions:
%e A368422 {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
%e A368422 {{1},{2}} {{1},{2,3}} {{1,2},{1,2}} {{1},{2,3,4,5}}
%e A368422 {{2},{1,2}} {{1},{2,3,4}} {{1,2},{3,4,5}}
%e A368422 {{1},{2},{3}} {{1,2},{3,4}} {{1,4},{2,3,4}}
%e A368422 {{1,3},{2,3}} {{2,3},{1,2,3}}
%e A368422 {{3},{1,2,3}} {{4},{1,2,3,4}}
%e A368422 {{1},{2},{3,4}} {{1},{2,3},{2,3}}
%e A368422 {{1},{3},{2,3}} {{1},{2},{3,4,5}}
%e A368422 {{1},{2},{3},{4}} {{1},{2,3},{4,5}}
%e A368422 {{1},{2,4},{3,4}}
%e A368422 {{1},{4},{2,3,4}}
%e A368422 {{2},{1,3},{2,3}}
%e A368422 {{2},{3},{1,2,3}}
%e A368422 {{3},{1,3},{2,3}}
%e A368422 {{4},{1,2},{3,4}}
%e A368422 {{1},{2},{3},{4,5}}
%e A368422 {{1},{2},{4},{3,4}}
%e A368422 {{1},{2},{3},{4},{5}}
%t A368422 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368422 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368422 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368422 Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]
%Y A368422 The case of unlabeled graphs is A134964, complement A140637.
%Y A368422 Set multipartitions have ranks A302478, cf. A073576.
%Y A368422 The case of labeled graphs is A133686, complement A367867.
%Y A368422 The complement without repeats is A368094 connected A368409.
%Y A368422 Without repeats we have A368095, connected A368410.
%Y A368422 The complement allowing repeats is A368097, ranks A355529.
%Y A368422 Allowing repeated elements gives A368098, ranks A368100.
%Y A368422 Factorizations of this type are counted by A368414, complement A368413.
%Y A368422 The complement is counted by A368421.
%Y A368422 A000110 counts set partitions, non-isomorphic A000041.
%Y A368422 A003465 counts covering set-systems, unlabeled A055621.
%Y A368422 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368422 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368422 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368422 Cf. A302545, A306005, A317533, A318360, A367903.
%K A368422 nonn,more
%O A368422 0,3
%A A368422 _Gus Wiseman_, Dec 26 2023