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A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

%I A330784 #4 Jan 04 2020 09:52:55
%S A330784 1,1,1,1,3,2,1,5,9,5,1,9,28,36,16,1,13,69,160,164,61,1,20,160,580,
%T A330784 1022,855,272,1,28,337,1837,4996,7072,4988,1385
%N A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.
%C A330784 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.
%F A330784 T(n,3) = A000041(n) - 2.
%F A330784 T(n,4) = A001970(n) - 3 * A000041(n) + 3.
%e A330784 Triangle begins:
%e A330784     1
%e A330784     1    1
%e A330784     1    3    2
%e A330784     1    5    9    5
%e A330784     1    9   28   36   16
%e A330784     1   13   69  160  164   61
%e A330784     1   20  160  580 1022  855  272
%e A330784     1   28  337 1837 4996 7072 4988 1385
%e A330784 Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
%e A330784   5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
%e A330784      {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
%e A330784      {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
%e A330784      {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
%e A330784      {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
%e A330784                 {{1,1},{1,2}}
%e A330784                 {{2},{1,1,1}}
%e A330784                 {{1},{1},{1,2}}
%e A330784                 {{1},{2},{1,1}}
%t A330784 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330784 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330784 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330784 Table[Length[Select[totm[ConstantArray[1,n]],Depth[#]==k&]],{n,2,6},{k,2,n}]
%Y A330784 Row sums are A318813.
%Y A330784 Column k = 3 is A007042.
%Y A330784 Column k = 4 is A001970(n) - 3*A000041(n) + 3.
%Y A330784 Column k = n is A000111.
%Y A330784 Row n is row prime(n) of A330727.
%Y A330784 Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.
%K A330784 nonn,more,tabl
%O A330784 2,5
%A A330784 _Gus Wiseman_, Jan 03 2020