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A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

%I A368727 #12 Jan 11 2024 20:04:20
%S A368727 1,1,2,2,5,6,15,21,49,82,184,341,766,1530,3428,7249,16394,36009,82492,
%T A368727 186485,433096,1001495,2358182,5554644,13255532,31718030,76656602,
%U A368727 185982207,454889643,1117496012,2764222322,6868902152,17172601190
%N A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.
%H A368727 Andrew Howroyd, <a href="/A368727/b368727.txt">Table of n, a(n) for n = 0..50</a>
%F A368727 Inverse Euler transform of A339888.
%e A368727 Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
%e A368727   {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
%e A368727        {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
%e A368727                           {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
%e A368727                           {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
%e A368727                           {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
%e A368727                                         {1}{1}{1}{1}{1}  {1}{2}{12}{12}
%e A368727                                                          {1}{2}{13}{23}
%e A368727                                                          {2}{2}{12}{12}
%e A368727                                                          {2}{2}{13}{23}
%e A368727                                                          {2}{3}{13}{23}
%e A368727                                                          {3}{3}{13}{23}
%e A368727                                                          {1}{1}{2}{2}{12}
%e A368727                                                          {1}{2}{2}{2}{12}
%e A368727                                                          {2}{2}{2}{2}{12}
%e A368727                                                          {1}{1}{1}{1}{1}{1}
%t A368727 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A368727 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368727 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 A368727 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368727 Table[Length[Union[brute /@ Select[mpm[n],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]
%Y A368727 For edges of any size we have A056156, with loops A007718.
%Y A368727 This is the connected case of A339888.
%Y A368727 Allowing loops {x,x} gives A368726, Euler transform A320663.
%Y A368727 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A368727 A007716 counts non-isomorphic multiset partitions, into pairs A007717.
%Y A368727 A062740 counts connected loop-graphs, unlabeled A054921.
%Y A368727 A320732 counts factorizations into primes or semiprimes, strict A339839.
%Y A368727 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A368727 Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.
%K A368727 nonn
%O A368727 0,3
%A A368727 _Gus Wiseman_, Jan 06 2024