A368410 - OEIS

%I #5 Dec 26 2023 08:31:57

%S 0,1,1,2,3,7,15,32,80,198,528

%N Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

%C 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.

%C 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.

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

%e Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 set-systems:

%e   {1}  {12}  {123}    {1234}    {12345}      {123456}

%e              {2}{12}  {13}{23}  {14}{234}    {125}{345}

%e                       {3}{123}  {23}{123}    {134}{234}

%e                                 {4}{1234}    {15}{2345}

%e                                 {2}{13}{23}  {34}{1234}

%e                                 {2}{3}{123}  {5}{12345}

%e                                 {3}{13}{23}  {1}{14}{234}

%e                                              {12}{13}{23}

%e                                              {1}{23}{123}

%e                                              {13}{24}{34}

%e                                              {14}{24}{34}

%e                                              {3}{14}{234}

%e                                              {3}{23}{123}

%e                                              {3}{4}{1234}

%e                                              {4}{14}{234}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];

%t brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];

%t 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]]]]]]]]];

%t Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

%Y For unlabeled graphs we have A005703, connected case of A134964.

%Y For labeled graphs we have A129271, connected case of A133686.

%Y The complement for labeled graphs is A140638, connected case of A367867.

%Y The complement without connectedness is A367903, ranks A367907.

%Y Without connectedness we have A368095, ranks A367906,

%Y Complement with repeats: A368097, connected case of A368411, ranks A355529.

%Y The complement is counted by A368409, connected case of A368094.

%Y With repeats allowed: A368412, connected case of A368098, ranks A368100.

%Y A000110 counts set-partitions, non-isomorphic A000041.

%Y A003465 counts covering set-systems, unlabeled A055621.

%Y A007716 counts non-isomorphic multiset partitions, connected A007718.

%Y A058891 counts set-systems, unlabeled A000612, connected A323818.

%Y A283877 counts non-isomorphic set-systems, connected A300913.

%Y Cf. A001055, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 25 2023