cp's OEIS Frontend

A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.

%I A368412 #8 Feb 07 2025 09:26:12
%S A368412 0,1,2,4,11,25,75,206,650,2049,6895
%N A368412 Number of non-isomorphic connected multiset partitions of weight n satisfying a strict version of the axiom of choice.
%C A368412 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 A368412 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 A368412 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A368412 Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
%e A368412   {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
%e A368412          {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
%e A368412                   {{1,2,3}}    {{1,2,2,2}}
%e A368412                   {{2},{1,2}}  {{1,2,3,3}}
%e A368412                                {{1,2,3,4}}
%e A368412                                {{1},{1,2,2}}
%e A368412                                {{1,2},{1,2}}
%e A368412                                {{1,2},{2,2}}
%e A368412                                {{1,3},{2,3}}
%e A368412                                {{2},{1,2,2}}
%e A368412                                {{3},{1,2,3}}
%t A368412 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368412 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368412 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368412 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 A368412 Table[Length[Union[brute /@ Select[mpm[n],Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]
%Y A368412 The case of labeled graphs is A129271, connected case of A133686.
%Y A368412 The complement for labeled graphs is A140638, connected case of A367867.
%Y A368412 This is the connected case of A368098, ranks A368100.
%Y A368412 Complement set-systems: A368409, connected case of A368094, ranks A367907.
%Y A368412 For set-systems we have A368410, connected case of A368095, ranks A367906.
%Y A368412 The complement is A368411, connected case of A368097, ranks A355529.
%Y A368412 A000110 counts set partitions, non-isomorphic A000041.
%Y A368412 A003465 counts covering set-systems, unlabeled A055621.
%Y A368412 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368412 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368412 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368412 Cf. A140637, A302545, A316983, A317533, A319616, A367902, A367905, A368187.
%K A368412 nonn,more
%O A368412 0,3
%A A368412 _Gus Wiseman_, Dec 26 2023