A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 4, 5, 0, 30, 2, 9, 9, 20, 3, 37, 6, 18, 16, 37, 20, 71, 12, 37, 40

Offset: 1

Author

Gus Wiseman, Jun 03 2019

Keywords

• nonn
• more

Comments

An integer partition is knapsack if every distinct submultiset has a different sum.

Examples

			The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.
		

Crossrefs

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.

Programs

Mathematica

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Union[#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]