cp's OEIS Frontend

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

%I A330726 #5 Jan 04 2020 09:52:40
%S A330726 1,1,2,3,7,17,54,199,869,4341,24514,154187
%N A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.
%C A330726 A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
%e A330726 The a(1) = 1 through a(5) = 17 multisystems (commas elided):
%e A330726   {1}  {2}   {3}        {4}               {5}
%e A330726        {11}  {12}       {22}              {23}
%e A330726              {{1}{11}}  {13}              {14}
%e A330726                         {{1}{12}}         {{1}{13}}
%e A330726                         {{2}{11}}         {{1}{22}}
%e A330726                         {{{1}}{{1}{11}}}  {{2}{12}}
%e A330726                         {{{11}}{{1}{1}}}  {{3}{11}}
%e A330726                                           {{{1}}{{1}{12}}}
%e A330726                                           {{{11}}{{1}{2}}}
%e A330726                                           {{{1}}{{2}{11}}}
%e A330726                                           {{{12}}{{1}{1}}}
%e A330726                                           {{{2}}{{1}{11}}}
%e A330726                                           {{{{1}}}{{{1}}{{1}{11}}}}
%e A330726                                           {{{{1}}}{{{11}}{{1}{1}}}}
%e A330726                                           {{{{1}{1}}}{{{1}}{{11}}}}
%e A330726                                           {{{{1}{11}}}{{{1}}{{1}}}}
%e A330726                                           {{{{11}}}{{{1}}{{1}{1}}}}
%t A330726 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330726 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330726 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330726 Table[Sum[Length[Select[totm[m],Depth[#]==If[Length[m]<=1,2,Length[m]]&]],{m,IntegerPartitions[n]}],{n,0,5}]
%Y A330726 The case with all atoms equal to 1 is A000111.
%Y A330726 The non-maximal version is A330679.
%Y A330726 A tree version is A320160.
%Y A330726 Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.
%K A330726 nonn,more
%O A330726 0,3
%A A330726 _Gus Wiseman_, Jan 03 2020