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A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

%I A320053 #9 Oct 05 2018 11:12:03
%S A320053 1,1,2,3,2,3,4,5,6,8,9,12,14
%N A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of  products of the parts of a multiset partition of y is distinct.
%e A320053 The sequence of spanning sum-product knapsack partitions begins:
%e A320053   0: ()
%e A320053   1: (1)
%e A320053   2: (2) (1,1)
%e A320053   3: (3) (2,1) (1,1,1)
%e A320053   4: (4) (3,1)
%e A320053   5: (5) (4,1) (3,2)
%e A320053   6: (6) (5,1) (4,2) (3,3)
%e A320053   7: (7) (6,1) (5,2) (4,3) (3,3,1)
%e A320053   8: (8) (7,1) (6,2) (5,3) (4,4) (3,3,2)
%e A320053   9: (9) (8,1) (7,2) (6,3) (5,4) (4,4,1) (4,3,2) (3,3,3)
%e A320053 A complete list of all sums of products covering the parts of (3,3,3,2) is:
%e A320053         (2*3*3*3) = 54
%e A320053       (2)+(3*3*3) = 29
%e A320053       (3)+(2*3*3) = 21
%e A320053       (2*3)+(3*3) = 15
%e A320053     (2)+(3)+(3*3) = 14
%e A320053     (3)+(3)+(2*3) = 12
%e A320053   (2)+(3)+(3)+(3) = 11
%e A320053 These are all distinct, so (3,3,3,2) is a spanning sum-product knapsack partition of 11.
%e A320053 An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).
%t A320053 sps[{}]:={{}};
%t A320053 sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320053 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320053 rtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,mps[q],{2}],{1}]]];
%t A320053 Table[Length[rtuks[n]],{n,8}]
%Y A320053 Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.
%Y A320053 Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.
%K A320053 nonn,more
%O A320053 0,3
%A A320053 _Gus Wiseman_, Oct 04 2018