A368094 - OEIS

A368094

Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

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OFFSET

0,7

COMMENTS

A set-system is a finite set of finite nonempty sets. The weight of a set-system 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.

LINKS

Table of n, a(n) for n=0..10.

Wikipedia, Axiom of choice.

EXAMPLE

Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:

{{1},{2},{3},{2,3}} {{1},{2},{1,3},{2,3}} {{1},{2},{1,2},{3,4,5}}

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

{{2},{3},{1,3},{2,3}} {{1},{4},{1,4},{2,3,4}}

{{3},{4},{1,2},{3,4}} {{2},{3},{2,3},{1,2,3}}

{{1},{2},{3},{4},{3,4}} {{3},{1,2},{1,3},{2,3}}

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

{{1},{2},{3},{2,4},{3,4}}

{{1},{2},{3},{4},{2,3,4}}

{{1},{3},{4},{2,4},{3,4}}

{{1},{4},{5},{2,3},{4,5}}

{{2},{3},{4},{1,2},{3,4}}

{{1},{2},{3},{4},{5},{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], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]], {n, 0, 8}]

CROSSREFS

The case of unlabeled graphs is A140637, complement A134964.

The case of labeled graphs is A367867, complement A133686.

The labeled version is A367903, ranks A367907.

The complement is counted by A368095, connected A368410.

Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.

Minimal multiset partitions of this type are ranked by A368187.

The connected case is A368409.

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

Allowing repeated edges gives 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, A326031, A330223, A330227, A367905.

Sequence in context: A229043 A185699 A301873 * A077918 A300534 A359189

Adjacent sequences: A368091 A368092 A368093 * A368095 A368096 A368097

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Dec 23 2023

STATUS

approved