A368097 - OEIS

%I #10 Dec 28 2023 11:32:05

%S 0,0,1,3,12,37,133,433,1516,5209,18555

%N Number of non-isomorphic multiset partitions of weight n contradicting 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(2) = 1 through a(4) = 12 multiset partitions:

%e   {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}

%e              {{1},{1},{1}}  {{1,1},{1,1}}

%e              {{1},{2},{2}}  {{1},{1},{1,1}}

%e                             {{1},{1},{2,2}}

%e                             {{1},{1},{2,3}}

%e                             {{1},{2},{1,2}}

%e                             {{1},{2},{2,2}}

%e                             {{2},{2},{1,2}}

%e                             {{1},{1},{1},{1}}

%e                             {{1},{1},{2},{2}}

%e                             {{1},{2},{2},{2}}

%e                             {{1},{2},{3},{3}}

%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 unlabeled graphs appears to be A140637, complement A134964.

%Y These multiset partitions have ranks A355529.

%Y The case of labeled graphs is A367867, complement A133686.

%Y Set-systems not of this type are A367902, ranks A367906.

%Y Set-systems of this type are A367903, ranks A367907.

%Y For set-systems we have A368094, complement A368095.

%Y The complement is A368098, ranks A368100, connected case A368412.

%Y Minimal multiset partitions of this type are ranked by A368187.

%Y The connected case is A368411.

%Y Factorizations of this type are counted by A368413, complement A368414.

%Y For set multipartitions we have A368421, complement A368422.

%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, A316983, A317533, A319616, A321405, A367905, A368409, A368410.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 25 2023