A368726 - OEIS

%I #15 Jan 11 2024 20:04:14

%S 1,1,3,3,8,10,26,38,93,161,381,732,1721,3566,8369,18316,43280,98401,

%T 234959,549628,1327726,3175670,7763500,18905703,46762513,115613599,

%U 289185492,724438500,1831398264,4641907993,11853385002,30365353560

%N Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

%H Andrew Howroyd, <a href="/A368726/b368726.txt">Table of n, a(n) for n = 0..50</a>

%F Inverse Euler transform of A320663.

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:

%e   {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}

%e          {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}

%e          {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}

%e                                    {{1,3},{2,3}}      {{2},{1,2},{2,2}}

%e                                    {{1},{1},{1,1}}    {{2},{1,3},{2,3}}

%e                                    {{1},{2},{1,2}}    {{3},{1,3},{2,3}}

%e                                    {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}

%e                                    {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}

%e                                                       {{2},{2},{2},{1,2}}

%e                                                       {{1},{1},{1},{1},{1}}

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

%t Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

%Y For edges of any size we have A007718.

%Y This is the connected case of A320663.

%Y The case of singletons and strict pairs is A368727, Euler transform A339888.

%Y A000085, A100861, A111924 count set partitions into singletons or pairs.

%Y A007716 counts non-isomorphic multiset partitions, into pairs A007717.

%Y A062740 counts connected loop-graphs, unlabeled A054921.

%Y A320732 counts factorizations into primes or semiprimes, strict A339839.

%Y A322661 counts covering loop-graphs, unlabeled A322700.

%Y Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 06 2024