OFFSET
0,4
COMMENTS
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.
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.
LINKS
Wikipedia, Axiom of choice.
EXAMPLE
Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 set multipartitions:
{{1},{1}} {{1},{1},{1}} {{1},{1},{2,3}} {{1},{1},{2,3,4}}
{{1},{2},{2}} {{1},{2},{1,2}} {{2},{1,2},{1,2}}
{{2},{2},{1,2}} {{3},{3},{1,2,3}}
{{1},{1},{1},{1}} {{1},{1},{1},{2,3}}
{{1},{1},{2},{2}} {{1},{1},{3},{2,3}}
{{1},{2},{2},{2}} {{1},{2},{2},{1,2}}
{{1},{2},{3},{3}} {{1},{2},{2},{3,4}}
{{1},{2},{3},{2,3}}
{{1},{3},{3},{2,3}}
{{2},{2},{2},{1,2}}
{{1},{1},{1},{1},{1}}
{{1},{1},{2},{2},{2}}
{{1},{2},{2},{2},{2}}
{{1},{2},{2},{3},{3}}
{{1},{2},{3},{3},{3}}
{{1},{2},{3},{4},{4}}
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], And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]=={}&]]], {n, 0, 6}]
CROSSREFS
The complement is counted by A368422.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 26 2023
STATUS
approved