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%I #5 Jul 12 2019 20:20:24
%S 1,1,3,11,51,259,1461,9133
%N Number of normal multiset partitions of weight n where each part has a different sum.
%C A multiset partition is normal if it covers an initial interval of positive integers.
%H Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e The a(0) = 1 through a(3) = 11 normal multiset partitions:
%e {} {{1}} {{1,1}} {{1,1,1}}
%e {{1,2}} {{1,1,2}}
%e {{1},{2}} {{1,2,2}}
%e {{1,2,3}}
%e {{1},{1,1}}
%e {{1},{1,2}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e {{2},{1,2}}
%e {{2},{1,3}}
%e {{1},{2},{3}}
%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 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Total/@#&]],{n,0,5}]
%Y Cf. A038041, A255906, A275780, A317583, A321469, A326517, A326518, A326520, A326521, A326535.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jul 12 2019
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