A368409 Number of non-isomorphic connected set-systems of weight n contradicting a strict version of the axiom of choice.

0, 0, 0, 0, 1, 0, 3, 5, 16, 41, 130

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(4) = 1 through a(8) = 16 set-systems:
  {1}{2}{12}  .  {1}{2}{13}{23}  {1}{3}{23}{123}    {1}{5}{15}{2345}
                 {1}{2}{3}{123}  {1}{4}{14}{234}    {2}{13}{23}{123}
                 {2}{3}{13}{23}  {2}{3}{23}{123}    {3}{13}{23}{123}
                                 {3}{12}{13}{23}    {3}{4}{34}{1234}
                                 {1}{2}{3}{13}{23}  {1}{2}{13}{24}{34}
                                                    {1}{2}{3}{14}{234}
                                                    {1}{2}{3}{23}{123}
                                                    {1}{2}{3}{4}{1234}
                                                    {1}{3}{4}{14}{234}
                                                    {2}{3}{12}{13}{23}
                                                    {2}{3}{13}{24}{34}
                                                    {2}{3}{14}{24}{34}
                                                    {2}{3}{4}{14}{234}
                                                    {2}{4}{13}{24}{34}
                                                    {3}{4}{13}{24}{34}
                                                    {3}{4}{14}{24}{34}

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]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]], {n, 0, 6}]

Crossrefs

For unlabeled graphs we have A140636, connected case of A140637.

For labeled graphs: A140638, connected case of A367867 (complement A133686).

This is the connected case of A368094.

The complement is A368410, connected case of A368095.

Allowing repeats: A368411, connected case of A368097, ranks A355529.

Complement with repeats: A368412, connected case of A368098, ranks A368100.

Allowing repeat edges only: connected case of 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. A134964, A302545, A306005, A317533, A321405, A326031, A367903, A367907, A368187, A368413.

Sequence in context: A226205 A300533 A221783 * A371868 A243321 A099101

Adjacent sequences: A368406 A368407 A368408 * A368410 A368411 A368412

Keyword

nonn,more

Author

Gus Wiseman, Dec 25 2023

Status

approved