A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

1, 1, 3, 7, 21, 54, 165, 477, 1501, 4736, 15652

Offset: 0

Author

Gus Wiseman, Dec 25 2023

Keywords

• nonn
• more

Comments

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.

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.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,2,2}}
                    {{1},{2,2}}    {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{2},{3}}  {{1,1},{2,2}}
                                   {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{1},{2},{3},{4}}
		

Links

• Wikipedia, Axiom of choice.

Crossrefs

The case of labeled graphs is A133686, complement A367867.

The case of unlabeled graphs is A134964, complement A140637 (apparently).

Set-systems of this type are A367902, ranks A367906, connected A368410.

The complimentary set-systems are A367903, ranks A367907, connected A368409.

For set-systems we have A368095, complement A368094.

The complement is A368097, ranks A355529.

These multiset partitions have ranks A368100.

The connected case is A368412, complement A368411.

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

For set multipartitions we have A368422, complement A368421.

A000110 counts set partitions, non-isomorphic A000041.

A003465 counts covering set-systems, unlabeled A055621.

A007716 counts non-isomorphic multiset partitions, connected A007718.

A058891 counts set-systems, unlabeled A000612, connected A323818.

A283877 counts non-isomorphic set-systems, connected A300913.

Cf. A302545, A306005, A316983, A317533, A319616, A330223, A330227, A368187.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]!={}&]]], {n,0,6}]