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A368422

Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.

1, 1, 2, 4, 9, 18, 43, 95, 233, 569 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,3`
	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, 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	`Table of n, a(n) for n=0..9.` `Wikipedia, Axiom of choice.`
	EXAMPLE	`Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 set multipartitions:` `{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}` `{{1},{2}} {{1},{2,3}} {{1,2},{1,2}} {{1},{2,3,4,5}}` `{{2},{1,2}} {{1},{2,3,4}} {{1,2},{3,4,5}}` `{{1},{2},{3}} {{1,2},{3,4}} {{1,4},{2,3,4}}` `{{1,3},{2,3}} {{2,3},{1,2,3}}` `{{3},{1,2,3}} {{4},{1,2,3,4}}` `{{1},{2},{3,4}} {{1},{2,3},{2,3}}` `{{1},{3},{2,3}} {{1},{2},{3,4,5}}` `{{1},{2},{3},{4}} {{1},{2,3},{4,5}}` `{{1},{2,4},{3,4}}` `{{1},{4},{2,3,4}}` `{{2},{1,3},{2,3}}` `{{2},{3},{1,2,3}}` `{{3},{1,3},{2,3}}` `{{4},{1,2},{3,4}}` `{{1},{2},{3},{4,5}}` `{{1},{2},{4},{3,4}}` `{{1},{2},{3},{4},{5}}`
	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 case of unlabeled graphs is A134964, complement A140637.` `Set multipartitions have ranks A302478, cf. A073576.` `The case of labeled graphs is A133686, complement A367867.` `The complement without repeats is A368094 connected A368409.` `Without repeats we have A368095, connected A368410.` `The complement allowing repeats is A368097, ranks A355529.` `Allowing repeated elements gives A368098, ranks A368100.` `Factorizations of this type are counted by A368414, complement A368413.` `The complement is counted by 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, A317533, A318360, A367903.` `Sequence in context: A032175 A000678 A362037 * A283877 A331850 A373808` `Adjacent sequences: A368419 A368420 A368421 * A368423 A368424 A368425`
	KEYWORD	`nonn,more`
	AUTHOR	`Gus Wiseman, Dec 26 2023`
	STATUS	`approved`

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Last modified August 21 06:47 EDT 2024. Contains 375345 sequences. (Running on oeis4.)