%I #8 Jan 20 2024 16:07:16

%S 1,1,2,4,12,40,180,936,5820,41288,331748,2968688,29307780,316273976,

%T 3704154568,46788812168,634037127612,9174782661984,141197140912208,

%U 2302765704401360,39671953757409256,719926077632193848,13726066030661998220,274313334040504957368

%N Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

%C 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.

%H Andrew Howroyd, <a href="/A330679/b330679.txt">Table of n, a(n) for n = 0..200</a>

%F a(n > 1) = 2 * A318813(n).

%e The a(0) = 1 through a(4) = 12 multisystems:

%e {} {1} {2} {3} {4}

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

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

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

%e {1,1,1,1}

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

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

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

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

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

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

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

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

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];

%t Table[Sum[Length[totm[m]],{m,IntegerPartitions[n]}],{n,0,5}]

%Y The case where the atoms are all 1's is A318813 = a(n)/2.

%Y The version where the atoms constitute a strongly normal multiset is A330475.

%Y The version where the atoms cover an initial interval is A330655.

%Y The maximum-depth version is A330726.

%Y Cf. A000041, A000111, A000669, A001970, A002846, A005121, A141268, A196545, A213427, A318812, A320160, A330474.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 31 2019

%E a(12) onwards from _Andrew Howroyd_, Jan 20 2024