cp's OEIS Frontend

A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

%I A368421 #7 Dec 27 2023 09:41:15
%S A368421 0,0,1,2,7,16,47,116,325,861
%N A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.
%C A368421 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 A368421 The axiom of choice says that, given any sequence of nonempty sets Y, 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 A368421 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368421 Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 set multipartitions:
%e A368421   {{1},{1}}  {{1},{1},{1}}  {{1},{1},{2,3}}    {{1},{1},{2,3,4}}
%e A368421              {{1},{2},{2}}  {{1},{2},{1,2}}    {{2},{1,2},{1,2}}
%e A368421                             {{2},{2},{1,2}}    {{3},{3},{1,2,3}}
%e A368421                             {{1},{1},{1},{1}}  {{1},{1},{1},{2,3}}
%e A368421                             {{1},{1},{2},{2}}  {{1},{1},{3},{2,3}}
%e A368421                             {{1},{2},{2},{2}}  {{1},{2},{2},{1,2}}
%e A368421                             {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
%e A368421                                                {{1},{2},{3},{2,3}}
%e A368421                                                {{1},{3},{3},{2,3}}
%e A368421                                                {{2},{2},{2},{1,2}}
%e A368421                                                {{1},{1},{1},{1},{1}}
%e A368421                                                {{1},{1},{2},{2},{2}}
%e A368421                                                {{1},{2},{2},{2},{2}}
%e A368421                                                {{1},{2},{2},{3},{3}}
%e A368421                                                {{1},{2},{3},{3},{3}}
%e A368421                                                {{1},{2},{3},{4},{4}}
%t A368421 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368421 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368421 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368421 Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]
%Y A368421 The case of unlabeled graphs is A140637, complement A134964.
%Y A368421 Set multipartitions have ranks A302478, cf. A073576.
%Y A368421 The case of labeled graphs is A367867, complement A133686.
%Y A368421 With distinct edges we have A368094 connected A368409.
%Y A368421 The complement with distinct edges is A368095, connected A368410.
%Y A368421 Allowing repeated elements gives A368097, ranks A355529.
%Y A368421 The complement allowing repeats is A368098, ranks A368100.
%Y A368421 Factorizations of this type are counted by A368413, complement A368414.
%Y A368421 The complement is counted by A368422.
%Y A368421 A000110 counts set partitions, non-isomorphic A000041.
%Y A368421 A003465 counts covering set-systems, unlabeled A055621.
%Y A368421 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368421 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368421 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368421 Cf. A302545, A306005, A316983, A317533, A318360, A367903, A367905, A367907.
%K A368421 nonn,more
%O A368421 0,4
%A A368421 _Gus Wiseman_, Dec 26 2023