proposed

approved

editing

proposed

The complimentary setSet-systems not of this type are A367902, ranks A367906, connected A368410.

Set-systems of this type are A367903, ranks A367907, connected A368409.

Minimal multiset partitions of this type are counted ranked by A368187.

Cf. A302545, A316983, A317533, A319616, A321405, A367905, A368409, A368410.

approved

editing

reviewed

approved

proposed

reviewed

editing

proposed

Set-systems of this type are counted by A367903, ranks A367907, connected A368409.

Cf. A302545, A306005, `A316983, A317533, `A319616, ~A321155, A321405, `A330223, `A330227, ~A367905.

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}]

allocated for Gus WisemanNumber of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.

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

0,4

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.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 multiset partitions:

{{1},{1}} {{1},{1,1}} {{1},{1,1,1}}

{{1},{1},{1}} {{1,1},{1,1}}

{{1},{2},{2}} {{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{1},{2,3}}

{{1},{2},{1,2}}

{{1},{2},{2,2}}

{{2},{2},{1,2}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{2},{2}}

{{1},{2},{3},{3}}

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}]

The case of unlabeled graphs appears to be A140637, complement A134964.

These multiset partitions have ranks A355529.

The case of labeled graphs is A367867, complement A133686.

The complimentary set-systems are A367902, ranks A367906, connected A368410.

Set-systems of this type are counted by A367903, ranks A367907, connected A368409.

For set-systems we have A368094, complement A368095.

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

Minimal multiset partitions of this type are counted by A368187.

The connected case is A368411.

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

For set multipartitions we have A368421, complement A368422.

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, ~A321155, A321405, `A330223, `A330227, ~A367905.

allocated

nonn,more

Gus Wiseman, Dec 25 2023

approved

editing

allocated for Gus Wiseman

allocated

approved