%I #7 Dec 26 2023 08:33:44
%S 1,1,3,7,21,54,165,477,1501,4736,15652
%N Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.
%C A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.
%C 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
%e {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
%e {{1,2}} {{1,2,2}} {{1,1,2,2}}
%e {{1},{2}} {{1,2,3}} {{1,2,2,2}}
%e {{1},{2,2}} {{1,2,3,3}}
%e {{1},{2,3}} {{1,2,3,4}}
%e {{2},{1,2}} {{1},{1,2,2}}
%e {{1},{2},{3}} {{1,1},{2,2}}
%e {{1,2},{1,2}}
%e {{1},{2,2,2}}
%e {{1,2},{2,2}}
%e {{1},{2,3,3}}
%e {{1,2},{3,3}}
%e {{1},{2,3,4}}
%e {{1,2},{3,4}}
%e {{1,3},{2,3}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1},{2},{3,3}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{1},{2},{3},{4}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]!={}&]]], {n,0,6}]
%Y The case of labeled graphs is A133686, complement A367867.
%Y The case of unlabeled graphs is A134964, complement A140637 (apparently).
%Y Set-systems of this type are A367902, ranks A367906, connected A368410.
%Y The complimentary set-systems are A367903, ranks A367907, connected A368409.
%Y For set-systems we have A368095, complement A368094.
%Y The complement is A368097, ranks A355529.
%Y These multiset partitions have ranks A368100.
%Y The connected case is A368412, complement A368411.
%Y Factorizations of this type are counted by A368414, complement A368413.
%Y For set multipartitions we have A368422, complement A368421.
%Y A000110 counts set partitions, non-isomorphic A000041.
%Y A003465 counts covering set-systems, unlabeled A055621.
%Y A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A283877 counts non-isomorphic set-systems, connected A300913.
%Y Cf. A302545, A306005, A316983, A317533, A319616, A330223, A330227, A368187.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Dec 25 2023